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In: Statistics and Probability

Let Z1, Z2, . . . , Zn be independent and identically distributed as standard normal...

Let Z1, Z2, . . . , Zn be independent and identically distributed as standard normal random variables. Prove the distribution of ni=1 Zi2 ∼ χ2n.

Thanks!

Expert Solution

The probability density function of distribution is as follows,

Moment generating function of distribution is as follows.

We substitute and so

We now calculate moment generating function of .

Probability mass function of standard normal distribution is given by

We substitute

Since, are independent.

By the uniqueness theorem of moment generating function, we can conclude that these two distribution are same an hence

orchestra answered 2 years ago

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