Exercise 8. Suppose X1, X2, , Xn are independent random variables with X; ~ N(Hi, o), 1

3- Show that, if the random variables X and Y are N(0,0,0,o',r) , then ept-2- 27(2-r²) 1 a) E{f,(y|x)}= 20 (2-r) 1 b) E{fx(x)fy(y)}=- 2no' 4-r

2. Let X and Y be two random variables. Suppose XY ~ Binomial (n, Y) where n is fixed and Y stands for the probability of success. Suppose the success probability Y is a random variable distributed as Beta(a, B). The PDF of Beta(a, B) is defined by the following a Beta distribution, Y f(u) = { Blaß)y'(1 – y)-1, 0 0, ß > 0. ) 0,B > 0, 0, otherwise, where B(a, B) = T(@)I(ß) T(a+B)· (1) Find E(X|Y) and V(X|Y). (2) Find E(X). (3) Find Cor(X,Y). Hint: E(X|Y) is a linear function of Y. (4) Find the conditional PDF of Y given X r. 3. Suppose X and Y ero

Let  X ~ Bernoulli (p). let Y be some discrete random variable. Know that Y is independent of X.     What is the condition of Y to satisify E(X+Y)=0 and V(X+Y)=2p?

Let X1,X2, .,Xn be i.i.d. random variables, where X; ~ Bernoulli(p). Define Y1 = X1X2, Y2 = X2X3, Yn-1 = Xn-1Xn, Y, = X,X1. If Y = Y1 + Y2 +……+ Yn, find 1. E[Y], 2. Var(Y).

Suppose the (cumulative) distribution function for a continuous random variable X is given by

Suppose X₁, X2, ..., X₁ are independent random variables and have CDF F(x). Find the CDF of Y = min{X₁, X₂,, Xn} and Z = max{X₁, X₂, ···‚ Xn}.

If we let X1, X2,...Xn be a random sample from the distribution N( μx = 8, σx = 4), then what would the distribution of (X-8)/(4/√n) be?   If we let X1, X2,...X10 be a random sample from the distribution N( μ, σ), then what is the value k so that P( (X-μ)/(σ/√10) < k ) = .05?

Part II: Sections 2.1 2.7 7. Suppose that X is a discrete random variable with pmf f(x) =D c.14-피 . 뉴- 21 for r=-3, 3, 6, where c is a constant. (a) Find c. Compute E ( ~LX +) (b) 3.

B4. Let X₁,... Xn ~ N(μ, o2) be independent random variables. 2 (a) From lectures we know (X=X) ²³. X₂ (b) Let n i=1 ~ x²(v). What is the value of v? n 1 *³ =, ²-₁, [(x₁ - x) ². Σ(x s n 1 i=1 Determine Var(s), that is, the variance of the sample variance. (c) Assume now that we have observed data ₁,...,n ER with sample variance s². Use the result from part (a) to find numbers an and b, such that [ans, bns] is an exact 95%-confidence interval for o².