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Two independent random variables, $X$ and $Y$, are uniformly distributed on the unit interval (-1,1).

  1. Determine the density for $U=min(X,Y)$ and for $W=max(X,Y)$
  2. Find the expectation for each variable $(U $ and $ W)$
  3. Find variance of each variable ($X,U,$ and $W)$
  4. Find the variance of $(U + W)$

I know that $U$ and $W$ are dependent on each other. I want to understand how to get the first question since if I understand that, I should be able to get the second one. I also think I will understand how to get variance of $U$ and $W$ but not for $X$. Thanks for whomever helps.

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Whenever you're asked to find the density function of a random variable, unless it's known, the safest way to find said probability density function (PDF) is to first find the cumulative distribution function (CDF) and then differentiate the CDF to obtain the PDF. The way you find the CDF of any random variable X is to compute P{X < x}.

In your case, you're asked to find the PDF of U=min(X,Y). Therefore, begin by finding the CDF of U. That is:

P{U < u} = P{min(X,Y) < u} = 1 - P{min(X,Y) > u}. Why are we using the last equality? Because if the smaller of X and Y is greater than u, then so is the larger one, so both are larger than u. You want to use the and operator to your advantage, since X and Y are independent. Therefore, you get the following:

P{U < u} = P{min(X,Y) < u} = 1 - P{min(X,Y) > u} = 1 - P{(X>u) and (Y>u)} = 1 - P{X>u}P{Y>u}, since X and Y are independent. At this point, it is very easy to compute those probabilities, since you are given the distribution of X and Y. ir7 already explained how to obtain the CDF of W. Cheers!

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Hint: Use the independence to get the cumulative distribution function: $$P(\max(X,Y) \leq z) = P(\{X \leq z\}\cap \{Y \leq z\}) = P(X \leq z) P(Y \leq z).$$ Differentiate it when you are done to get the probability density function.

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Hints for 1:

  • What is the probability that $X \gt u$?
  • What is the probability that $Y \gt u$?
  • What is the probability that $X \gt u$ and $Y \gt u$?
  • What is the probability that $U \le u$?
  • What is the density for $U$?
  • How would you find the density for $W$ in a similar way?
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  • $\begingroup$ What is the variable $u$? $\endgroup$ – YoungGrasshopper Feb 13 '14 at 22:54
  • $\begingroup$ $u$ is a number between $-1$ and $1$, and you are trying to find the density of $U$ at $u$. It is not random. $\endgroup$ – Henry Feb 13 '14 at 22:58
  • $\begingroup$ I'm just not getting it. I need step by step for the first one. I know that if something is uniformly distributed it looks like a rectangle. I just need help with the first problem $\endgroup$ – YoungGrasshopper Feb 13 '14 at 23:13

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