Question

ppolt & Taluom Variable has edf. F(x), then the probability that X lies in the interval [a, b) is Question 2. 30 pt.) Single-choice questions 1) Suppose A and B are independent events, then)is incorrect. P(AIB) = P(A) B P(AB)- P(A) D PCA u B) = P(A) + P(B) e P(A B)-P(A)P(B) 2) Suppose X-Bin(10,0.3) and Y-Bin(15,0.3), they are independent, thenis incorrect. oX+Y-Bin(25,0.3) GX+Y-N(7.5,5.25) D VarX)VarCr) 3) Suppose X N(0,1) and YN(2,4), they are independent, then is incorrecet X + Y~N(2, 5) B P(Y <2)>0.5 X-Y-N(-2,5) Varx)Var(r) 4) Suppose x N(H,0) and YN(H20), they are independent, then in the following statementsis incorrect. P(X < μ.) = 0.5 Suppose X-N(u1,of) and Y-N(u203), they are independent, if p(X-μ|く1) > PCIY-μ| < 1), then ( 5) )ǐs correct. Notice: In order to receive full credit your answer must be complete, legible and correct. Show all of your work, and give adequate explanations.

Answer 1

If A and B are independent then p(A Intersection B) = P(A) P(B) So, P (A/B) = P(A Intersection b)/P(B) = P(A) P(B)/P(B) = P(A) P(A Union B) = P(A) + P(B) - P(A) P(B) P(A/B^-) = P(A Intersection B^-)/P(B^-) = P(A) P(B^-)/P (B^-) = P (A) Hence, option (d) is incorrect 2) X ~ Bin (10, 0.3) , y ~ Bin (115, 0.3) Then by property x + y ~ (10 + 15, 0.3) ~ (25,0.3) X + Y ~ Bin (25,0.3) E(x) = 10 times 0.3 = 3, E(Y) = 15 times 0.3 = 4.5 E(Y) > E(X) Var(X) = 10 times 0.3 times 0.7 = 2.1 = 3.15 Var (y) > Var (X) Hence, option (c) is incorrect. 3) X ~ N (0,1) and Y ~ N(2,4) Then by property x + y ~ N(0 + 2, 1 + 4) X + Y ~ N (2,5) P(Y < 2) = P(Y < mu) = 0.5 Var(Y) = 4, Var (x) = 1 Var(y) > Var(x) X - Y ~ N (0 -2, 1 + 4) ~N (-2, 5) Hence, Option (B) is incorrect. 4) X ~ N (mu_1, sigma^2_1) &