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2. Suppose that X Binom(n,p) such that n>1 and 0 <p<1. Show that E[(x + 1)-1...
8. Let f (x) e, 0 > 0; x> 0 (1 1 +e (a) Show that f(x) is a probability density function (b) Find P(X> x) (c) Find the failure rate function of X
5. Let A € Mnxn(C) with characteristic polynomial p(x) = cxºII-1(d; – x) and li + 0, Vi, a E Z>o. Show that if dim(ker(A))+k=n, then A= C2 for some complex matrix C.
/2 for n E N. Use the Monotone Convergent (2) Suppose that o E R and xn (1 Theorem to prove that xn >1 as n -> 0.
8) Assume that X ~ N(μ = 4,02-1). Find c >0 such that P(-c 〈 X 〈 c) Find P(2 〈 X 〈 6) a. 0.95 b.
2. Suppose Xi ~ N(8,02) where θ > 0. (a) Show that s--(x, Σ¡! xi) is a sufficient statistic of θ where X is the sample mean. (b) Is S minimal sufficient? (c) Can you find a non-constant function g(.) such that g(S) is an ancillary statistic?
| Prove that for n e N, n > 0, we have 1 x 1!+ 2 x 2!+... tnx n! = (n + 1)! - 1.
2. Prove that if n > 1, then 1(1!) + 2(2!) + ... + n(n!) = (n + 1)! - 1.
Find the area under the graph of g over [-2, 3] g(x) = -x? +5 when x 50 g(x) = x + 5 when x > 0
Exercise 6. Show that if f(x) > 0 for all x e [a, b] and f is integrable, then Sfdx > 0.
IF Let x(t) Show that e 20" σ>0, and let (o) be the Fourier transform of x(t) .