Question

If a random variable x has a chi-square distribution with

K degrees of freedom, find the moment generating function of
x.
A. Mx(t)=(1-2t)-K/2 B. Mx(t)=(1+2t)-k/2 C. Mx(t)=e-kt/2 D. None of the above

Answer 1

The moment generating function (MGF) for a random variable x with a chi-square distribution and K degrees of freedom is Mx(t) = (1 - 2t)^(-K/2), so the correct answer is A.

Step-by-step explanation:

If a random variable x has a chi-square distribution with K degrees of freedom, the moment generating function (MGF) can be found using the properties of the chi-square distribution. The chi-square distribution is the distribution of a sum of the squares of K independent standard normal random variables.

To find the MGF of a chi-square distribution, recall that the MGF of a single squared standard normal random variable (Z²) is (1 - 2t)-1/2. Since the chi-square random variable x is the sum of K such independent square normal variables, the MGF of the chi-square distribution is the product of their individual MGFs. Therefore, the MGF for a chi-square distribution with K degrees of freedom is: Mx(t) = (1 - 2t)-K/2

Among the options provided, the correct answer is A. Mx(t) = (1 - 2t)-K/2.