Question

You are told that E(X )= 2 and var(X) = 3. Variable Y = 3X+2 and variable Z = 1/2X−5.

(a) Compute E(X² ) and E(Y²).

Answer 1

To find the z-value for y = 4 when Y is normally distributed with a mean of 2 and a variance of 1, we use the z-score formula which yields a z-value of 2.

Step-by-step explanation:

The student is dealing with a question on the topic of expected values and variances of random variables. Specifically, they are given the means and variances of normally distributed random variables X and Y, and they have to determine a new value of the standard normal variable z corresponding to a given value of y. To solve the problem involving the normal distribution of Y with mean 2 and variance 1, and to find the corresponding z-value for y = 4, we use the z-score formula:

Z = (Y - μ) / σ

Where μ is the mean and σ is the standard deviation. For Y ~ N(2,1), the mean μ is 2 and the standard deviation is the square root of the variance, which is 1. Thus, the z-score for y = 4 is:

Z = (4 - 2) / 1 = 2

Therefore, the value of z when y = 4 is 2.