X, ~ Nu, ) and Y, ~ Nu, 1). Find the M LES ont Let X be the number of independen

Question

X, ~ Nu, ) and Y, ~ Nu, 1). Find the M LES ont Let X be the number of independent trials of some component until it fails, wh the probability of failure on each trial, we record the exact number of trials, 14 where -ps r. r; otherwise we record Y = r + 1, where r is a fixed positive integer (a) Show that the discrete pdf of Y is y=r+1 (b) Let Y..., Y, be a random sample from /: p). Find the MLE of p. Hint :.ry; p) = dy, pri, where dy, p)-1-pify-l, ,rand dr + 1;p)-1.It tous that dy; p) = (1-p)" where n is the number of observed y, that are less than r (15/ Let X ~ BIN(n, p) and p = X/n. (a) Find a constant c so that E[cp(1 P)] p(1 -p. (b) Find an unbiased estimator of Var(X). (c) Consider a random sample of size N from BIN(n, p). Find unbiased estimators of o and Var(X) based on the random sample. 16.) (Truncated Poisson.) Let X ~ POI(4), and suppose we cannot observe X 0, so the observed random variable, Y, has discrete pdf f(y; ) = y!(1-e-") y=1,2, .. We desire to estimate P[X ased (hut unrsonablel

Explanation / Answer

15)
a)
E(p^) = p
E(p^ (1-p^)/n) = pq/n        {q = 1-p}
hence E(p^* (1-p^)) = p(1-p)
hence
c = 1

b)
Var(X) = np (1-p)
since E(p^(1-p^)) = p (1-p)
hence
unbiased estimator of Var(X) = n p^ (1-p^)

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