Statistics 100B
EXERCISE 1
Instructor: Nicolas Christou
University of California, Los Angeles Department of Statistics
Quiz 5
2U Let U ∼ N(0,1), V ∼ χn, and U and V are independent. Let t = V
n
Suppose Q1,…,Qk are independent. Let Q1 ∼ χ2p1(θ1),…,Qk ∼ χ2pk(θk), where p1,…,pk are the degrees of free- dom and θ1, . . . , θk are the non-centrality parameters. Find the mean and variance of Y = Q1 + . . . + Qk.
EXERCISE 3
Let X1, X2, . . . , Xn i.i.d. exponential random variables with parameter λ. Is 1 unbiased estimator of λ? X ̄
and W = V. Find the joint pdf of t and W and then integrate the joint w.r.t. to W to show that the probability density function of the t distribution with
df = n degrees of freedom is
−n+1
Γ(n+1) t2 2
f(t)=√ 2 1+ , −∞