University of California, Los Angeles Department of Statistics
Instructor: Nicolas Christou
Quiz 6
A probability problem.
Let’s Make a Deal! A player is asked to choose one of three doors. Behind one of the doors there is a prize. Suppose the player chose door 1. The host of the game who knows where the prize is opens door 3 and the player sees that there is no prize behind this door. Then the host asks the player: Would you like to switch to door 2 or stay with door 1? Define the following events: Hi : Host opens door i and Di : Prize is behind door i. Find P(D2|H3) and P(D1|H3) to show that switching has higher probability.
EXERCISE 2
Let X1, X2, . . . , Xn be independent exponential random variables with mean iθ. For example, E(X1) = θ, E(X2) = 2θ, etc. Suppose an estimate of of θ is θˆ= n (Xi ).
i=1 ni
a. Find the distribution of θˆ. b. Find E[θˆ−1].
c. Find the MSE of cθˆ−1 as an estimator of θ−1, and find c that minimizes that MSE. EXERCISE 3
Answer the following questions:
a. Let X1,X2,…,Xn be i.i.d. random variables with Xi ∼ Poisson(λ). Show that θ1 = n i=1 Xi(Xi − 1) is
unbiased estimate of λ2.
efficient estimator of σ2? First find the information using two different methods, for example, the variance of
the score function and −E ∂2lnf(x;θ) . ∂θ2
c. Let X1, . . . , Xn i.i.d. random variables with Xi ∼ N(μ, σ). Is X ̄ a consistent estimator of μ? What if var(Xi) = σ2 and cov(Xi,Xj) = ρσ2? Is X ̄ a consistent estimator of μ?
Statistics 100B
EXERCISE 1
ˆ 1n
b. Let X1, X2, . . . , Xn i.i.d. normal random variables with Xi ∼ N(0, σ). Consider σˆ2 = 1 n Xi2. Is σˆ2 an
n i=1
1