Suppose that Y_(1) and Y_(2) are independent, standard normal random variables. Find the density

Jaylen Dudley 2022-09-12 Answered
Suppose that Y 1 and Y 2 are independent, standard normal random variables. Find the density
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Answers (1)

Dwayne Small
Answered 2022-09-13 Author has 12 answers
This distribution is χ 2 with v = 2. So both our random variables have m ( t ) = ( 1 2 t ) 1 / 2 , t < 1 2 Note that when U=A+B where A and B are independent random variables we get m U ( t ) = m a ( t ) m B ( t ) By this logic we get
m U ( t ) = ( 1 w t ) 1
Note that this is the moment generating function for an exponential random variable with λ = 1 2 . So we get the following density function.
f U ( u ) = 1 2 e u / 2 , u 0

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