9. IfX is a r.), distributed as N(μ, σ2), find the value of c (in terms of μ and σ) for which P(Xc 2-9P(X > c).
-. Show that if x ~ Unif(a, β) and μ-E(X) (so μ-(a + β)/2), then for integers r > 0, 0 r odd 12
1. Given a continuous random number x, with the probability density P(x) = A exp(-2x) for all x > 0, find the value of A and the probability that x > 1.
Prove that is an integer for all n > 0.
Exhibit 9-1 n = 36 X 24.6 Ho: μ s 20 Hai μ>20 12 The p-value is O a. 2.1 O b..0107 c. .0214 d. .5107
Compute E[X] if X has a density function given by 0 Otherwise If X is a normal random variable with parameters μ = 10 and σ = 6, compute 1. P(X > 5) 2. P (4 < X < 16) 3. P (X < 8) 4. P (IXI> 16)
2. Assume X is a random variable following from N(μ, σ2), where σ > 0. (a) Write down the pdf of X (b) Compute E(X2). (b) Define YFind the distribution of Y.
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
Problem 4. The median of a PDF fx(x) is defined as the number a for which P(X s a)-P(X > a)-1/2. Find the median of a Gaussian PDF N(μ; σ2).
2. Suppose that X Binom(n,p) such that n>1 and 0 <p<1. Show that E[(x + 1)-1 = _(1 – p)p+1 – 1 p(n + 1)