Question

Consider 256 independent random variable X1, X2,... , X256. These random variables are also identically distributed, meaning that their variance is the same. Assume that Var(X1)= 16. The mean of these random variables is X. What is the SD(X)?

Answer 1

To calculate the standard deviation of the mean (SD(X)) for 256 independent and identically distributed random variables with a variance of 16 each, you first find the SD of one variable (4), then divide by the square root of the number of variables (16) to get an SD(X) of 0.25.

Step-by-step explanation:

The question involves understanding the concept of standard deviation (SD) for a distribution of independent and identically distributed random variables. Given that there are 256 such random variables with the same variance, and the variance of one of them (Var(X1)) is 16, we are looking to find the SD of the mean of these random variables.

To find the standard deviation of the mean, also known as the standard error, we use the formula SD(X) = SD(X1) / √(n), where n is the number of random variables. Since we are given Var(X1) = 16, we first find the SD of a single random variable X1, which is SD(X1) = √(Var(X1)) = √(16) = 4. Knowing the number of variables is 256, we then find the SD of the mean: SD(X) = 4 / √(256) = 4 / 16 = 0.25.