Question

Please answer A.6.6.:

The previous two questions mentioned above are included below:

A.6.6. We mentioned in class that the Gamma(, 2) distribution when k is a positive integer is called the Chi-square distribution with k degrees of freedom. From the previous two problems, find the mean, variance, and MGF of the Chi-square distribution with k degrees of freedom. A.6.5. In class we showed that if X ~ Gamma(α, β) then E (X) = aß and uar(X) = αβ2 by using the normalizing constant idea. Your job: use the MGF from to compute the mean and variance. A.6.4. Let X ~ Gamma(a, 3). Show that the MGF of X is M(t) (1-St)-α.

Answer 1

Date A.。.. DLúw.ng the mean Chi-squara with dsarees ot freedom kis ane as gaa disibuti ahametests 2 2 Thus, uue iLL fast desive gamma distúbuío Paf of gamn a distributi oX and dividing by d/x 0 The integra ant becomes the pdf of the is bing integhated on ths whole

E[X] = う Substituting tha Values of the ch- . gua 1/2. n exacty tha Sanma mannun usin ral vaiance coms l.Va.tx ] = E [x2]-EEJ32-2k ED can be calcula ed as :- 2-

We UǐL'Subshtute the values of chi- Ihis gves and Substifutin = (d-大) TC So

We hawe used the definition 04 tha amma functien in the Last stap Thus MGF for Chi-suare distribution 2