Let Y be a random variable from N(0, r−1 ) and r be a random variable from Gamma (ν/2, ν/2), respectively. Note that the

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Let Y be a random variable from N(0, r−1 ) and r be a random
variable from Gamma (ν/2, ν/2), respectively. Note that the pdf of
Gamma (ν/2, ν/2) random variable is p(r) = (ν/2)ν/2 Γ(ν/2) r ν/2−1
e −νr/2 . (a) Show that the marginal distribution of Y is Student’s
t distribution with degree of freedom (df) ν. Note that the pdf of
distribution with df ν is p(y) = ν −1/2Γ((ν + 1)/2) √ πΓ(ν/2) 1 + y
2 ν −(ν+1)/2 . (b) Present an algorithm to generate tν random
variable and generate 10000 random numbers using above result. Then
calculate mean of the 10000 random numbers and compare true mean
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