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Let $X$ be a random digit $(0,1,2, \ldots, 9$ are equally likely) and let $Y$ be a random digit not equal to$X .$ That is, the nine digits other than $X$ are equally likely for $Y .$(a) Determine $p_{X}(x), p_{Y X}(y | x), p_{X, Y}(x, y)$(b) Determine a formula for $E(Y | X=x)$

(a) $p_x(x) = 1/10 $for$ x = 0, 1, ..., 9; p(y|x) = 1/9 for y = 0, ..., 9 $and$ y 6\ne x; p(x, y) = 1/90 for x, y = 0,1, ..., 9 $and$ y 6\ne x (b) 5 x/9$

Intro Stats / AP Statistics

Chapter 4

Joint Probability Distributions and Their Applications

Section 1

Jointly Distributed Random Variables

Probability Topics

The Normal Distribution

Foster J.

October 29, 2021

A pizza place has two phones. On each phone the waiting time until the first call is exponentially distributed with mean one minute. Each phone is not influenced by the other. Let X be the shorter of the two waiting times and let Y be the longer. It can b

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One single-digit number is…

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were given a random variable X, which represents a random digit and were given why, which is a random digit not equal to X. So the nine digits other than X are equally likely. Why in part A. Or has to determine the marginal probability density function of X, the conditional probability density function of why given X and the joint Probability density function of X and Y well, we have that all tended. It's equally likely so marginal. P x of X is going people too. 1/10 for ex Equalling 01 or any other digit up to nine. Now we have that conditional probability of why given X of why given X is going to be equal to 1/9 or why Equalling 01 up 39. But why not Equalling X? Of course, Yeah, this is because any than nine being digits are equally likely finally combining. We get that probability. Joint probability density function of X and Y p X y is well, the marginal probability p X connects times the conditional probability p y given x of why given X and now we have that this is going to be well, 1/10 times 1/9 which is 1/90. A zloty, as we have that X and Y, can be equal to 01 digit through nine. But why has to not equal to X since for all pairs x y such that with X and Y are different digits next in Part B or asked to determine a formula for expected value of why, given X equals X, we have that this is equal to We're going to sob over the possible values of why. So we have some over all. Why equaling 01 through nine asl Ong as why does not equal to X of Why times the conditional probability. P Why given x of y given X and we have that above. In part A. This is going to be 1/9 times, some from why equals euro nine or by is not equal to X of why this is simply 1/9 times zero plus one plus and so on up until we get to nine. And then we have to also subtract from the some of the the 10 digits the digit X because we don't want you don't want to count that. So this adds up to 1/9 times the some from 09 This is going to be nine times 10 over to which is nine times five, which is 45 minus x. In simplifying we get this is equal to five minus X over nine. She's the expected value of why given the X equals X.

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