Question

let x1,x2,...x5 be independent and let each have the same distribution with mean 10 and variance 4 . find e(x1 x2 ... x5) and var(x1 x2 ... x5)

Answer 1

The expected value of the product of the independent variables is 100,000, and the variance of the product is 1,024, given each has a mean of 10 and variance of 4.

Step-by-step explanation:

The calculation of E(X1 X2 ... X5) involves finding the expected value of the product of independent random variables each with a mean of 10. Since they are independent, the expected value of their product is the product of their expected values, so:

E(X1 X2 ... X5) = E(X1)E(X2)...E(X5) = 10 * 10 * 10 * 10 * 10 = 100,000.

To find Var(X1 X2 ... X5), since the variables are independent, the variance of their product is the product of their variances, each raised to the power of 2 (due to the properties of variance for independent variables), so:

Var(X1 X2 ... X5) = [Var(X1)]^5 = (4)^5 = 1,024.

Note:

If the question intended to find the variance of the sum instead of the product, the process would differ. The variance of the sum of independent random variables is the sum of their variances:

Var(X1 + X2 + ... + X5) = Var(X1) + Var(X2) + ... + Var(X5) = 4 + 4 + 4 + 4 + 4 = 20 since each X has a variance of 4.