Question

1)Let X1, ..., Xn be independent standard normal random variables, we know that X2 1 + ... + X2 n follows the chi-squared distribution of n degrees of freedom. Find the third moment of the the chi-squared distribution of 2 degrees of freedom.

2) Suppose that, on average, 1 person in 1000 makes a numerical error in preparing his or her income tax return. If 10,000 returns are selected at random and examined, find the probability that 6 or 7 of them contain an error

Answer 1

Using the definition of Moment Generating Function

[We have replaced ]

[By the definition of Gamma function]

Again by the definition of c, we obtain

According to the problem n = 2

So the MGF is =

The same can be expanded as

Comparing with

or,