CS1A Gamma distribution

[Rishabh Joshi]

How to solve this one ?
Let X1, X2, X3,…,Xn be a random sample of claim amounts which follow gamma distribution
with a parameter α = 5 and unknown parameter λ.
i) Specify the distribution of ∑Xi
with parameters. Hence also state the distribution of
2nλXhat.
ii) A random sample of 10 such claims has a mean of 100. Pick the 95% confidence interval
for λ.

 

[Rishabh Joshi]

How to solve this one ?
Let X1, X2, X3,…,Xn be a random sample of claim amounts which follow gamma distribution
with a parameter α = 5 and unknown parameter λ.
i) Specify the distribution of ∑Xi
with parameters. Hence also state the distribution of
2nλXhat.
ii) A random sample of 10 such claims has a mean of 100. Pick the 95% confidence interval
for λ.

Solutions- (But it's not well explained ), Please if anyone can explain
i) Sum of Xi follows Gamma distribution with parameters 5n, λ
If Y ~ gamma (α, λ) then 2 λ Y ~ chi squared distribution with degree of freedom 2α

Hence 2nλXhat follows chi squared distribution with df 10n.
ii) Confidence Interval is (0.03711, 0.06480)

 

[John Lee]

i) This is covered in Chapter 4 (Joint Distributions) in the notes using MGFs. Set \(Y=\sum X_i\) and find the MGF of Y.
ii) Presumably, you mean xhat is \(\bar X\) in which case \(n \bar X\) is equal to Y.
Then you're just proving the \(2 \lambda X \sim \chi ^2\) result which uses the results from Chapter 3 (GFs) in the notes.

This kind of question has been asked numerous times in the CT3 exams (eg A11 Q5, S10 Q8, S08 Q2, A08 Q7, A01 Q15)