Question

Please answer A.6.6.:

A.6.6. We mentioned in class that the Gamma(, 2) distribution when k is a positive integer is called the Chi-square distribut

The previous two questions mentioned above are included below:

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. Yo

A.6.4. Let X ~ Gamma(a, 3). Show that the MGF of X is M(t) (1-St)-α.

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.
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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 desiE[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]-EEJWe UǐLSubshtute the values of chi- Ihis gves and Substifutin = (d-大) TC SoWe hawe used the definition 04 tha amma functien in the Last stap Thus MGF for Chi-suare distribution 2

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