$X_i$ Execution plan - reading more records than in table. \sum_{k=1}^i \frac{1}{n-k-t+1} \right]_{t=0}\\ &=\frac{n!}{(i-1)!(n-i)! Mean of maximum of exponential random variables (independent but not identical) Ask Question Asked 4 years, . The joint pdf then transforms as, \begin{equation*} is incorrect, because the event Connect and share knowledge within a single location that is structured and easy to search. Asking for help, clarification, or responding to other answers. So We omit the case where two or more are equal since this occurs with probability zero: includes the first, second, and fifth inequalities, of which only the first two also satisfies It's just expanding the product: $(1-a)(1-b)=1-a-b+ab$, Mean of maximum of exponential random variables (independent but not identical), Mobile app infrastructure being decommissioned. Is Z an exponential random variable with parameter + ? Are certain conferences or fields "allocated" to certain universities? For example, my answer includes the probability that the elevator breaks down at $5$ AM and then breaks down again at $6$ AM. \begin{aligned}[b] $$ Show activity on this post. Browse other questions tagged, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site, Learn more about Stack Overflow the company. and so the probability is Is a potential juror protected for what they say during jury selection? Of course, the minimum of these exponential distributions has distribution: and X i is the minimum variable with probability i / . MIT, Apache, GNU, etc.) The continuous random variable, say X is said to have an exponential distribution, if it has the . Let Assume a day begins at midnight. is then The only way I've thought of verifying your first step is by noting $P(M > m | L = l) = P(M > m | L = l, X > Y)$ and then you see $P(M > m | L = l) = P(X > m | Y = l, X > l) = P(X > m | X > l)$ by independence of $X$ and $Y$. I think you're thinking in terms of 'largest of two expos' as the prior and 'the smaller one has value $ l$' as the additional info, which would make sense given the form of the left-hand-side of the equation, but wasn't how I was thinking about it. $$ $T_A < T_C$ Exponential Distribution Formula. \begin{aligned}[b] Given the order statistics $x_{(1)} \le x_{(2)} \le\le x_{(n)}$, Z_i = (n-i+1)(X_i - X_{i-1}) \qquad \longrightarrow \qquad X_i = \frac{Z_i}{n-i+1} + X_{i-1} Let $X_i$, $i=1,2$, denote the waiting time until the first event to occur in the process $N_i$. How many rectangles can be observed in the grid? $$\Pr\{Z\le 2\} = 1-\exp(-20/24).$$. Stack Exchange network consists of 182 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Traditional English pronunciation of "dives"? \sum_{i=1}^n x_i = \sum_{i=1}^n z_i $T_B < T_A < T_C$ The number of breakdowns of the first elevator in a day has a I meant 'You have an RV in front of you and you are told it's the larger of two iids and the smaller has value $l$. T_B < T_C < T_A \\ . Using this formula for expectation of positive random variables in terms of the survival function and expanding the product in your formula for the cdf, Consider n independent random variables X i exp ( i) for i = 1, , n. Let = i = 1 n i. Why was video, audio and picture compression the poorest when storage space was the costliest? \{X\wedge Y>t\} = \{X>t\}\cap\{Y>t\}, \end{aligned} The CDF : $ \mathbf{F_{X_{max}}}(x)= \prod_{k=1}^{K} (1-exp(-\lambda_k x)) $ and Use MathJax to format equations. $T_A < T_C$ Cross Validated is a question and answer site for people interested in statistics, machine learning, data analysis, data mining, and data visualization. In only the first two cases is $=1$ Thanks for contributing an answer to Cross Validated! To subscribe to this RSS feed, copy and paste this URL into your RSS reader. Why should you not leave the inputs of unused gates floating with 74LS series logic? what is $\mathbb{E}(Z)$? The Maximum and Minimum of Two IID Random Variables Suppose that X 1 and X 2 are independent and identically distributed (iid) continuous random variables. , and the event The best answers are voted up and rise to the top, Not the answer you're looking for? $$ A different approach is that we can view the order statistic as a sum statistic. Using exponential random graph models and data on eight Swiss wetlands, a qualitative meta-regression analysis of the results reveals that the three qualities of actor-issue interdependencies influence collaboration patterns between actors. . $P(T_A < min(T_B,T_C)) = \frac{1}{1+1+1} = \frac{1}{3}$. We also deal with the. I am looking for the the mean of the maximum of N independent but not identical exponential random variables. Then you can take advantage of the fact that a minimimum of expontially distributed variables is also expontially distributed. I'm just translating the additional info to 'it is bigger than $l$.' E[X_{i:n}] &= E\left[\frac{Z_{1:n}}{n} + \frac{Z_{2:n}}{n-1} ++ \frac{Z_{i:n}}{n-i+1}\right] and Is there an industry-specific reason that many characters in martial arts anime announce the name of their attacks? $$ What is the probability that the minimum of X and Y is below z? The first part of the sentence was the main logical step, not the independence part. By independent, we mean that PfX 1 2A;X 2 2Bg= PfX 1 2AgPfX 2 2Bg for any A R and B R. By identically distributed we mean that X 1 and X 2 each have \mathbb P( X_kt)\mathbb P(Y>t)\\ So I define a RV as X \begin{aligned}[b] $$ I didn't use the memoryless property in the 1st 2 parts so I suspect it will come in for this last part. $$, Minimum of Two Exponential Random Variables, Minimum of two exponentially distributed random variables, Understanding the distribution of the minimum of two exponential random variables. Suppose $x_1, x_2, , x_n$ are i.i.d. It only takes a minute to sign up. \end{equation*}. \end{align}, $X\wedge Y\sim\mathrm{Expo}(\lambda+\mu)$, $$ One would speak here not of the minimum of two exponential distributions, but of the minimum of two exponentially distributed random variables . \\&=\int_0^\infty1-\prod_{i=1}^n(1-e^{-\lambda_i x})dx Expected value of the Max of three exponential random variables, Pdf of sum of exponential random variables, Difference of two exponential distribution, What is the expected value of the maximum of two independent exponentially distributed random variables? To learn more, see our tips on writing great answers. $n$ Is the minimum of all Elevator $A$ breaks down with an Exponential Distribution of $4$ per day, while elevator $B$ breaks down with an exponential distribution of $6$ per day. Answer: That it is larger than $l$.' $T_A,T_B, T_C$ Connect and share knowledge within a single location that is structured and easy to search. Thus, let $Z = \min(X,Y)$, so $P(Z < 2) = 1-e^{-20} = 1$ approximately. . $X_1,\ldots,X_n$ What mathematical algebra explains sequence of circular shifts on rows and columns of a matrix? What is this political cartoon by Bob Moran titled "Amnesty" about? Covalent and Ionic bonds with Semi-metals, Is an athlete's heart rate after exercise greater than a non-athlete. Why is HIV associated with weight loss/being underweight? it is easy to find out that . $(b)$ To find the expected location of the leftmost point, we find the pdf (probability distribution function) of min (X, Y) and calculate the expectation using the pdf. $T_C < T_A < T_B$ Hence, $\min \{X_1,X_2\} \sim {\rm Exp}(\lambda_1 + \lambda_2)$. Let X,Y, and Z be independent exponential random variables with an average of 1. We omit the case where two or more are equal since this occurs with probability zero: $$T_A < T_B < T_C \\ \begin{aligned}[b] includes the case And that was very useful - thanks! So the short of the story is that Z is an exponential random variable with parameter 1 + 2, i.e., E(Z) = 1=( 1 + 2). The Jacobian of the transformation turns out to be $n!$ (see pg 101 of referenced paper). $1/6$ To analytically study how skewness affects its direction-finding performance, the hybrid Cramr-Rao bound (HCRB) of the directions-of-arrival . \\&=\sum_{S\subseteq\{1,2,\dots,n\}}(-1)^{|S|} \int_0^\infty e^{-x\sum_{j\in S}\lambda_j}dx My Problem: Now, for the sake of rigor and clarity, consider the full pdf of the ordered statistic for a general integer $i ; 1 {n > 1}]. $X$ and $Y$ are. How actually can you perform the trick with the "illusion of the party distracting the dragon" like they did it in Vox Machina (animated series)? I want to find \{X\wedge Y>t\} = \{X>t\}\cap\{Y>t\}, I think the step $P(M>m|L=l)=P(X>m|X>l)$ was clever and I probably wouldn't have come up with that on my own. The distribution function of Given a set of $n$ exponentially distributed i.i.d variables $X_i \sim EXP(1)$ the expected value of an ordered statistic $X_{i:n}$ is found in a straighforward fashion with the method of moments which gives the expected value as, \begin{equation*} Also. Among all continuous probability distributions with support [0, ) and mean , the exponential distribution with = 1/ has the largest differential entropy.In other words, it is the maximum entropy probability distribution for a random variate X which is greater than or equal to zero and for which E[X] is fixed. Let M m i n ( X, Y), where X, Y . Solution 2: Note that the rates are $4$ breakdowns per $24$ hours and $6$ breakdowns per $24$ hours. How can I calculate the number of permutations of an irregular rubik's cube? Thanks a lot. $T_B < T_A < T_C$ Why are UK Prime Ministers educated at Oxford, not Cambridge? How to split a page into four areas in tex. Is this homebrew Nystul's Magic Mask spell balanced? $$ To subscribe to this RSS feed, copy and paste this URL into your RSS reader. Stack Exchange network consists of 182 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. So Z is an exponential random variable with parameter + . Let W=max (Y,Z) and T=min (X,W). f(x_i) &= \frac{n!}{(i-1)!(n-i)! For part (b), if Do we ever see a hobbit use their natural ability to disappear? where the outer sum is over all non-empty subsets $S$ of $\{1,2,\dots,n\}$ and $|S|$ denotes the number of elements of $S$. How does DNS work when it comes to addresses after slash? I was reasoning conditionally. So although it is true that, $$T_A < \min(T_B, T_C) \equiv (T_A < T_B) \cap (T_A < T_C),$$, it is For small n you can now calculate moments of X m a x by integration, It's senseless to ask for a different answer just because the right answer isn't expressed in a way you like! for $n=2$, we have The process $N$ is a Poisson process with rate $\lambda_1 + \lambda_2$. $T_C < T_A < T_B$ then $_i$ Your verification is correct but it isn't what I 'meant'. And the rate of the next bus arriving should be the minimum of X. Unless the two random variables are independent you can say nothing about there joint distribution based on the knowledge of the marginal distributions. E[X] &= \frac{n!}{(i-1)!(n-i)! The biaxial velocity sensor comprises two nominally perpendicular particle velocity sensors and a collocated pressure sensor. Movie about scientist trying to find evidence of soul. $\begingroup$ Another way is exploiting $\max(X,Y)=X+Y-\min(X,Y)$. Let $X$ = the time between two consecutive breakdowns for elevator A and $Y$ = the time between two consecutive breakdowns for elevator B. \end{aligned} How to go about finding a Thesis advisor for Master degree, Prove If a b (mod n) and c d (mod n), then a + c b + d (mod n). [4 Points] Show that the minimum of two independent exponential random variables with parameters \( \lambda \) and \( \mu \), respectively, is an exponential random variable with parameter \( \lambda+\mu \). \begin{align} However, suppose I am given the fact that X a is the minimum . $$, $$ T_C < T_B < T_A$$. \end{equation*}. By induction, if Can FOSS software licenses (e.g. = satisfied. , where \end{aligned}$$. But it is possible to obtain the answer through elementary means, beginning from definitions. Why are there contradicting price diagrams for the same ETF? rev2022.11.7.43014. and so on. The best answers are voted up and rise to the top, Not the answer you're looking for? What is the expectation of $Z$, i.e. Is this meat that I was told was brisket in Barcelona the same as U.S. brisket? $$ Will it have a bad influence on getting a student visa? \end{aligned} ). &=\int_0^\infty P(X_\text{max}>x)dx (3.19a) (3.19b) A plot of the PDF and the CDF of an exponential random variable is shown in Figure 3.9. Let $X_1 ~ \sim \text{Exp}(\lambda_1)$ and $X_2 \sim \text{Exp}(\lambda_2)$ be two independent exponentially distributed random variables. E X_\text{max} = \frac1{\lambda_1}, T_B < T_A < T_C \\ $k$ I'm not following. When the migration is complete, you will access your Teams at stackoverflowteams.com, and they will no longer appear in the left sidebar on stackoverflow.com. Estimation of $\lambda$, $\mu$, and $\sigma^2$ given observations of $Z=X+Y$, $X\sim\text{Poi}(\lambda)$, $Y\sim\mathcal N(\mu,\sigma^2)$, Probability that sum of independent uniform variables is less than 1, Sum of identically distributed but not independent Bernoulli's is non-uniform, Sum of random variables uniformly distributed (0,1) and (0,2), Expected value of maximum of two random variables from uniform distribution, Expectation of Minimum of $n$ i.i.d. One would speak here not of the minimum of two exponential distributions, but of the minimum of two exponentially distributed random variables. $\{X_k0$? What is rate of emission of heat from a body at space? }\left[\frac{\partial}{\partial t}\int [1-e^{-x_i}]^{i-1}[e^{-x_i}]^{(n-i+1-t)x_i}\right]_{t=0}\\ What is the probability that the first breakdown occurs before $2$ AM? and respectively . I found the CDF and the pdf but I couldn't compute the integral to find the mean of the maximum. In general you get for the $m$-th order statistic (of $n$ exponential distributed variables) the expectation: $$E[X_{(k)}] = \sum_{k=1}^m \frac{1}{n+1-k} $$. What is the probability of bus line k arriving first? are @drhab Oops, yes I did forget to mention that. Chat with a Tutor. $T_A < T_B$ different bus lines arrive. Can an adult sue someone who violated them as a child? Why plants and animals are so different even though they come from the same ancestors? We have $P(X>z) = e^{-z}$ since $X$ is a standard exponential, so $Z$ is a standard exponential. $$ Assume that at a bus stop, \end{equation*}. By clicking Accept all cookies, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy. Then $X_i \sim {\rm Exp}(\lambda_i)$, with $X_1$ and $X_2$ independent. Stack Overflow for Teams is moving to its own domain! Thanks for contributing an answer to Cross Validated! Mobile app infrastructure being decommissioned, Expected value of maximum of $n$ iid exponential random variables, Maximum Likelihood Estimator of the exponential function parameter based on Order Statistics, How to find maximum likelihood of multiple exponential distributions with different parameter values. What is the rate of the next bus arriving? &= \sum_{k=1}^i \frac{1}{n-t+1} Why is that first step true? That initial "$1$" in the integrand is thorny, because its integral diverges, so we cannot separate it out. Making statements based on opinion; back them up with references or personal experience. $P(T_A < min(T_B,T_C))$. @Lovsovs, yes I did intend that. Classic "Order Statistics" problem: Find the probability density function of the "Maximum and Minimum of Two Random Variables in terms of their joint probab. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level. $$ Transcribed image text: (a) Let X 1,X 2,,X n be independent exponential random variables with rate parameters 1,,n respectively. Let W=max (Y,Z) and T=min (X,W). Since probabilities of independent events multiply, $$\Pr(x_{(n)} \le x) = \left(1 - e^{-x}\right)^n.$$, One well-known formula for the expectation of a positive random variable with distribution function $F$ is the integral of $1-F$ from $0$ to $\infty.$ (Take the usual integral for the expectation and integrate by parts.) Z:= \bigwedge_{i=1}^n X_i \sim \mathrm{Expo}\left(\sum_{i=1}^n \lambda_i\right). Note that the \end{equation*}. Below I've given a formula for the cumulative distribution function (CDF) of the maximum of n independent exponentials (which, of course, is one way to specify a distribution); if you want the density, you can differentiate it. Protecting Threads on a thru-axle dropout. $$F_{x(1)}(x)= 1 - \Big[1-F_{x}(x)\Big]^n = 1- \Big[1-(1-e^{-x})\Big]^n=1-e^{-nx}$$ Z_k^- := \bigwedge_{i=1,i\ne k}^n X_i, Z:= \bigwedge_{i=1}^n X_i \sim \mathrm{Expo}\left(\sum_{i=1}^n \lambda_i\right). & = \sum_{k=1}^i \frac{1}{n-t+1} Is there a keyboard shortcut to save edited layers from the digitize toolbar in QGIS? Note that the rates are $4$ breakdowns per $24$ hours and $6$ breakdowns per $24$ hours. I was trying to perform this, but the integral is $\int_0^\infty x n (1-e^{-x})^{n-1}e^{-x}dx$, and by Taylor expansion, $1-e^{-x} = x - \frac{x^2}{2}+ \frac{x^3}{6} - \frac{x^4}{24} +$, which is not obvious that its n-1th power is an explicit term. $\endgroup$ For example, the amount of money spent by the customer on one trip to the supermarket follows an exponential distribution. Now assume that the and are independent and distributed with c.d.f. We are looking, then, to compute, $$E_n = E\left[x_{(n)}\right] = \int_0^\infty 1 - \left(1 - e^{-x}\right)^n\,\mathrm{d}x$$. I also want to add I asked some friends who know probability better than I do, and they suggested order statistics. f_{X_1,X_2,,X_n} = n!e^{-\sum_{i=1}^n x_i } \quad \longrightarrow \quad e^{-\sum_{i=1}^n z_i } Then. random variable of exponential distribution $Exp(1)$, i.e., $f(x)=e^{-x}, x\gt0$. The event that bus Then $\min(X,Y)$ is exponentially distributed with rate $10$. Show convergence of the first order statistic of independent uniform$(0,n)$ distributed random variables, Mean and variance of the maximum of a random number of Uniform variables, Solving a marginalization integral involving exponential distributions, Mean and Variance of Continuous Random Variable, Wikipedia Proof About Minimum of Exponential Random Variables, Find UMVUE of difference of parameters of two exponential distribution random variables. Z_k^- := \bigwedge_{i=1,i\ne k}^n X_i, $T_A < T_C$ &= \mathbb P(X>t)\mathbb P(Y>t)\\ How do you find the minimum of two exponential random variables? Why are standard frequentist hypotheses so uninteresting? because the events Why was video, audio and picture compression the poorest when storage space was the costliest? From that the conditional PDF can be computed to be $f_{M|L}(m|l) = e^{-(m-l)}1_{m\ge l}$ which can be integrated to give $P(M>m|L=l) = e^{-(m-l)}1_{m\ge l}$. $T_A, T_B, T_C$ $t>0$ This question hasn't been solved yet Ask an expert Ask an expert Ask an expert done loading. . Light bulb as limit, to what is current limited to? What was the significance of the word "ordinary" in "lords of appeal in ordinary"? &= e^{-(\lambda+\mu)t}, What do you call an episode that is not closely related to the main plot? If we take the maximum of 1 or 2 or 3 's each randomly drawn from the interval 0 to 1, we would expect the largest of them to be a bit above , the expected value for a single uniform random variable, but we wouldn't expect to get values that are extremely close to 1 like .9. $\lambda_1,\ldots,\lambda_n$ Did find rhyme with joined in the 18th century? How does the lack of memory property affect the exponential distribution? To learn more, see our tips on writing great answers. Is there a keyboard shortcut to save edited layers from the digitize toolbar in QGIS? X_i \sim \frac{Z_1}{n} + \frac{Z_2}{n-1} ++ \frac{Z_i}{n-i+1} Let's think about how $M$ is distributed conditionally on $L=l$. Your formal argument is correct but $M = \max(X,Y)$ is not distributed exponentially. $\lambda$ This . Stack Overflow for Teams is moving to its own domain! \\&=\sum_{S\subseteq\{1,2,\dots,n\}}(-1)^{|S|-1} \frac1{\sum_{j\in S}\lambda_j}, Name for phenomenon in which attempting to solve a problem locally can seemingly fail because they absorb the problem from elsewhere? Also E [ min ( X 1, X 2) + max ( X 1, X 2)] = E [ X 1 + X 2] = 1 + 1 . T_A < T_C < T_B \\ Durability of fabric glued to wood/plastic. are iid exponential distributions, they must be observed in one of the following orders, each of which is equally likely with probability Taking the derivative, How does DNS work when it comes to addresses after slash? Handling unprepared students as a Teaching Assistant. You are welcome. $Z$ independent. Which finite projective planes can have a symmetric incidence matrix? $$\Pr[T_A < \min(T_B, T_C)] = \Pr[T_A < T_B]\Pr[T_A < T_C]$$ F i ( x i) = 1 2 + 1 2 Erf [ ( x i i) / ( i 2], the cumulative distribution of the maximum is given by. . Another way is exploiting $\max(X,Y)=X+Y-\min(X,Y)$. from which it follows that parameter $$, \begin{align} My profession is written "Unemployed" on my passport. are independent exponentially distributed random variables with respective parameters , we have Order statistics (e.g., minimum) of infinite collection of chi-square variates? collaboration ties are the dependent variable, and different qualities of actor-issue paths are the . are not independent. But it is possible to obtain the answer through elementary means, beginning from definitions. \end{aligned} $1-e^{-x\sum_i \lambda_i}$ Stack Exchange network consists of 182 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Make use of: $$\mathbb EZ=\int_0^{\infty}P(Z>z)dz$$, and of course:$$P(Z>z)=P(X>z)+P(Y>z)-P(X>z\wedge Y>z)$$, By independence of $X,Y$ this results in:$$P(Z>z)=P(X>z)+P(Y>z)-P(X>z)P(Y>z)$$, Site design / logo 2022 Stack Exchange Inc; user contributions licensed under CC BY-SA. Due to real-world imperfections in manufacturing or setup errors, the two axes may suffer from perpendicularity losses. $$F_{x(n)}(x) = \Big[F_{x}(x)\Big]^n = (1-e^{-x})^n$$ The pdf is : $ \mathbf{f_{X_{max}}}(x)= \sum_{k=1}^{K}\lambda_k exp(-\lambda_k x) \prod_{q=1,q\neq k}^{K} (1-exp(-\lambda_q x)) $ Will it have a bad influence on getting a student visa? What's the best way to roleplay a Beholder shooting with its many rays at a Major Image illusion? The exponential random variable has a probability density function and cumulative distribution function given (for any b > 0) by. Min, Max, and Exponential. $$ The answer referenced in the comments is great, because it is based on straightforward probabilistic thinking. \begin{aligned}[b] You can see this explicitly from the table above: &= e^{-\lambda t}e^{-\mu t}\\ What have you been told about the RV in front of you? What is the ratio distribution of a spacing and the sample mean? If a) CDF of exponential distribution is 1ex where . Because E [ min ( X 1, X 2)] = 1 + , we get E [ max ( X 1, X 2)] = 1 + 1 1 + . . \end{equation*}. $$ T_C < T_A < T_B \\ Expectation of the maximum of two exponential random variables [closed], Mobile app infrastructure being decommissioned, Expected value of maximum of two random variables from uniform distribution, Expectation of three exponential random variables in a queue, Random sum of random exponential variables, Distribution of sum of exponential variables with different parameters, Finding Independent exponential random variables, Sum of exponential random variables with different parameters - followup, Sum of exponential random variables over their indices, Maximum of N iid random random variables with Gumbel distribution, Expected Value of the Maximum of 3 Independent Exponential Random Variables, Exponential random variables independency. If you need to use results from parts a-d, you dont Let V = max { X, Y }. T_B < T_C < T_A \\ $$ Cross Validated is a question and answer site for people interested in statistics, machine learning, data analysis, data mining, and data visualization. Should I avoid attending certain conferences. For independently distributed x i 's, each with cumulative distribution. . You (most probably) forgot to mention that $X,Y$ are independent. \end{aligned} test the hypothesis that the mean working hours is 16 hours against the hypothesis that . The exponential random variable can be either more small values or fewer larger variables. $X_i$ uniform random variables, Basic question about using the chi-square table, How do you prove Cov $\left( \bar{X} , X_i, Finding the distribution of the sum of three independent uniform random variables, Sql aggregate function in dbms code example, Javascript change dropdown with jquery code example, Javascript regex for strong password code example, Most common angular interview questions code example, Cpp multiple definition of function code example, File copy ubuntu terminal cmd code example, Python matplotlib histogram bin color code example, Shipping for specific user woocommerce code example, The minimum of two independent exponential random variables with parameters and is also exponential with parameter + . is exponentially distributed with rate Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. and Poisson How do you find the minimum number of variables greater than X. How to prove that minimum of two exponential random variables is another exponential random variable? \begin{aligned}[b] Rubik's Cube Stage 6 -- show bottom two layers are preserved by $ R^{-1}FR^{-1}BBRF^{-1}R^{-1}BBRRU^{-1} $. Sorry this is not an answer, but a question to @Jarle for some clarification I understand that there was a conversion between F(x) and S(x) between the two lines. They follow no reliable plan and the Following the answer on the link I gave above$$\mathbb{E}[X_{(n)}]=\sum_{i=1}^n \dfrac{1}{i}$$.
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maximum of two exponential random variables
