1. ## Independent Random Variables

3. Suppose we roll a dice n times. Define the random variable Xi to be equal to 1 if the
ith roll results in an even number, and 0 otherwise. Define the the random variable Yi to be
equal to 1 if the ith roll results in the number 6, and zero otherwise.
(a) Are X1 and Y1 independent random variables?
(b) Find the distribution of Z1 = X1 + Y1.
(c) Suppose n is 500. What is the approximate distribution of
1/500 (500)sigma(i=1)Xi +1/500 (500)sigma(i=1)Yi

2. Originally Posted by farshyfaroo
3. Suppose we roll a dice n times. Define the random variable Xi to be equal to 1 if the
ith roll results in an even number, and 0 otherwise. Define the the random variable Yi to be
equal to 1 if the ith roll results in the number 6, and zero otherwise.
(a) Are X1 and Y1 independent random variables?
$\displaystyle X_1$ and $\displaystyle Y_1$ are independednt if:

$\displaystyle P( (X_1=a) \wedge (Y_1=b))=P(X_1=a)P(Y_1=b)$

for all $\displaystyle a$ and $\displaystyle b$ in the appropriate sample spaces. Let $\displaystyle a=b=1$, does this hold?

(b) Find the distribution of Z1 = X1 + Y1.
The possible values of Z_1 are 0, 1, 2 calculate their probabilities.

(c) Suppose n is 500. What is the approximate distribution of
1/500 (500)sigma(i=1)Xi +1/500 (500)sigma(i=1)Yi
$\displaystyle U=\frac{1}{500}\sum_{i=1}^{500} X_i+\frac{1}{500}\sum_{i=1}^{500} Y_i=\frac{1}{500}\sum_{i=1}^{500} Z_i$

Now the $\displaystyle Z_i$'s are iid so the $\displaystyle U$ is the mean of a sample of size $\displaystyle 500$ for a sample from the same distribution as $\displaystyle Z_1$ ... so is approximatly normally distributed with mean and variance ...

CB