Exercise 8. Suppose X1, X2, , Xn are independent random variables with X; ~ N(Hi, o), 1
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- Let X be a random variable with pdff(x) = 4x^3 if 0 < x < 1 and zero otherwise. Use thecumulative (CDF) technique to determine the pdf of each of the following random variables: 1) Y=X^4, 2) W=e^(-x) 3) Z=1-e^(-x) 4) U=X(1-X)Suppose that X₁, X2, X3 are mutually independent random variables with the respective moment = (-1) ². generating functions, Mx, (t) = et², Mx₂ (t) = ²t + 3t², and Mx, (t) = (2
- 5. Suppose random observations Y₁ and Y2 are independent each distributed N(0, 1). Also let the following random Y² + Y22 variable be defined as s² = Show that 2s2~ 2p(x) = {ax^2 + c { 09 X, and X, are independent random variables each distributed as N (0, 1). Show that the MGF of (X,-X) and (X, + X,) is e'i+5.
- The joint cdf of a random variable X and Y is given by -ax Fxy(x, y) = {} ¹)(1-e²b) X,Y 0 (a) Find the marginal cdfs of X and Y. (b) Show that X and Y are independent. / = x≥0, y ≥0, a, b>0 otherwiseLet X1, X2, ..., Xn be independent and identically distributed random variables with N ~ (0, 1) and let Y; = X²,i = 1, ..., n. 2(Y1+Y2+Y3) 3(Y4+Y5) Find the distribution of W =The discrete random variable T has the p.m.f P (T=m)=2^m -1 / 3^m , m=0,1,...,m,... Show that E(T)=3 hence find the Var(T)
- 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-rIf X1 and X2 are independent random variables with distribution given by P[Xi] = −1 = P[Xi = 1] = 1/2 for i = 1, 2, then are X1 and X1X2 independent?b) Let Z₁ = X-XN (0,1), and W₁ dx YHY~N(0,1), for i = 1,2,3,...,10, then: dy i) State, with parameter(s), the probability distribution of the statistic, T = - 54 1² ii) Find the mean and variance of the statistic T = Σ},wp? Σ1,2,3 iii) Calculate the probability that a statistic T = Z₁ + W₁ is at most 4. iv) Find the value of ẞ such that P(T> B) = 0.01, where T = Σ₁Z₁² + ₁ W₁².