EXERCISE 1. Suppose Xi's are iid Negative Binomial(3,1/4) (a) Compute P(X1 < 5); (b) approximate P(21.9...
Suppose Xi's are independent and identically distributed Negative Binomial(3, 1/4): compute P(X1 <= 5).
1. Suppose that X, X, X, are iid Berwulli(p),0 <p<1. Let U. - x Show that, U, can be approximated by the N (np, np(1-P) distribution, for large n and fixed <p<1. 2. Suppose that X1, X3, X. are iid N ( 0°). Where and a both assumed to be unknown. Let @ -( a). Find jointly sufficient statistics for .
Let X1,X2 be two independent
exponential random variables with λ=1, compute the
P(X1+X2<t) using the joint density function. And let Z be gamma
random variable with parameters (2,1). Compute the probability that
P(Z < t). And what you can find by comparing P(X1+X2<t) and
P(Z < t)? And compare P(X1+X2+X3<t) Xi iid
(independent and identically distributed) ~Exp(1) and P(Z < t)
Z~Gamma(3,1) (You don’t have to compute)
(Hint: You can use the fact that Γ(2)=1,
Γ(3)=2)
Problem 2[10 points] Let...
4. Suppose Yi, Yn are iid randonn variables with E(X) = μ, Var(y)-σ2 < oo. For large n, find the approximate distribution of p = n Σηι Yi, Be sure to name any theorems you used.
Suppose that X1,X2, ,Xn are iid N(μ, σ2), where both parameters are unknown. Derive the likelihood ratio test (LRT) of Ho : σ2 < σ1 versus Ho : σ2 > σ.. (a) Argue that a LRT will reject Ho when w(x)S2 2 0 is large and find the critical value to confer a size α test. (b) Derive the power function of the LRT
(4) Suppose that {X;}-1 iid random variables from a Binomial distribution Bin(m, p). Using your answer in (3) obtain an approximate 99% confidence interval for the pa- rameter p based on the MLE. Explain how you would estimate the Fisher information matrix.
5. Compute the following binomial probabilities directly from the formula for b(x:n, p) a. 6(3:8, 35) b. b3< X < 5) when n=7 and p = .6 c. b(15 x) when n = 9 and p = .1
Consider a binomial experiment with n- 12 and p0.2 a. Compute f(0) (to 4 decimals). f(0) b. Compute f (8) (to 4 decimals). f(8) c. Compute P(x < 2) (to 4 decimals) Pa 2) d. Compute P1 (to 4 decimals). e. Compute E(z) (to 1 decimal). E(x) f. Compute Var(z) and σ. Var(x) (to 2 decimals) to 2 decimals) f. Compute the probability of six occurrences in three time periods (to 4 decimals).
4. Compute each probability or quantile 1 (а) X~ N(3, 0.0225), P(X < 3.25). (b) X~ N(52, 49), Р(X > 60). (с) X~ N(3.7, 4.55), Р(3.0 < х < 10.0). (d) XN(25, 36). Find the first and third quartiles for X.
5.2.5 (Example 5.2.6 Continued) Suppose thatXY are iid having the following common distribution. PC,-1-cip.i-1. 2. 3. and 2 <p < 3 Here. c c(p) (> 0) is such that Σ | P(X,-i) = 1.. Is there a real number a = a(p) such that Xn → a as n → 00, for all fixed 2 <p < 3? FYI: Example 5.2.6 In order to appreciate the importance of Khinchine's WLLN (Theorem 5.2.3). let us consider a sequence of iid random...