Question

Let X be a discrete random variable with 1 P(X = 1) = P(X = 2) = P(X = 3) = P(X= 4) = Then given X = x, we roll a fair 4-sided die 3 times. (The 4-sided die is equally likely to come up a 1, 2, 3, or 4). Let y be the number of times we roll a 1. (a) Find E[Y|X]. Hint: Remember E[Y | X] is a random variable, so X will be part of the answer. (Or if you prefer, you can list out the different cases.) (b) Find E[Y]

Answer 1

The probability of rolling a 1 on any given roll of a 4 sided die is 1/4=0.25

We roll the die X=x times. Let Y be the number of times with roll a 1. We can say that Y has a Binomial distribution with parameters, number of trials (number of times the dies is rolled) n=x and success probability (The probability of rolling a 1) p=0.25

That is the distribution of Y given X=x is

$Y\mid X\sim \text{Binomial}(X,0.25)$

a) The conditional expectation of Y given X is (using the formula for binomial distribution)

$E(Y\mid X)=np=X\times 0.25=0.25X$

ans:

$E(Y\mid X)=0.25X$

b) The pmf of X in tabular format is

x	P(x)
1	1/4
2	1/8
3	1/2
4	1/8

The expected value of X is

E(X) = ΣτΡ() =1x1/4 + 2x1/8 + 3x1/2 + 4x1/8 2,50

The unconditional expectation of Y is

E[Y] = ELE( YX)] using the law of iterated expectation = E(0.25X] substituting E( YX) = 0.25X = 0.25E(X) using the result E(aX +b) = aE(X) + b = 0.25 x 2.5 = 0.625

ans:

$\begin{align*} E[Y]=0.625 \end{align*}$