3. If X and Y are independent uniform (0, 1) random variables, show that 2 for a > 0 (a + 1)(a + 2) E[|X – Y|ª] -
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![3. If X and Y are independent uniform (0, 1) random variables, show that
2
(a + 1)(a + 2)
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- 1) If X,Y are two random variables which are NOT independent, then which of the following relations is always true? a. E(X +Y) > E(X)+E(Y) b. E(X +Y) = E(X)+E(Y) c. E(X +Y) < E(X)+E(Y) d. None of theseQ7. Let X; (i = 1,.,n) be a random sample from the N(u, o²) population with unknown parameters u and o?. Then we know that T, = E-1(X - X)²/(n – 1) is an unbiased estimator -xi-1 . Now consider two other estimators T, and T3 for o? where of a? and further, (n-1)T T2 = E(X – X)*/n and T3 = E(X – X)²/n + 1). %3D What is the variance of T,? Find out the variances and biases of T2 and T3 and thus obtain their m.s.e.'s. Show that among the three estimators T,, T2 and T3 the one with minimum m.s.e. is T3.Suppose that X, Y, and Z are random variables with X - Bin(16, 1/4), Y ~ Geom(1/2), and Z ~ Poisson(5). Suppose further that X and Z are independent but that X and Y are not independent. Match up the following quantities. E(4Y – 2Z) Drag answer here 23 Var(2X + Y) 8 Drag answer here Var(X – Z) -2 Drag answer here E(Y² + Z®) Cannot be determined Drag answer here 36 Var(X+ 2Z) Drag answer here
- Q.3. Compute the probability that: (a) their sum is odd; (b) their product is even. Three distinct integers are chosen at random from the first 15 positive integers.1. (Independence). (i) Recall that given events A, B,C, we have that P(AUBUC) = P(A)+ P(B)+ P(C) – P(An B) – P(AnC)– P(BnC) + P(AN BnC). (4.1) Prove that whenever events A, B,C are mutually independent, we have that P(AUBUC) = 1– P(A^)P(B^)P(C*) = 1– (1– P(A)) · (1 – P(B)) · (1 – P(C)) (and the probability in question can be calculated considerably faster than with the use of the formula (4.1)). (ii) The probability that A hits a target is 2/5, the probability that B hits it is 5/9, and the probability that C hits the target is 3/7. Use (i) to find the probability that the target will be hit if A, B, and C each shoot at the target. (iii) Suppose that the probability that a soldier firing his personal weapon hits an enemy warplane is p > 0. Show then, arguing as in (i), that the probability that the warplane is hit at least once when n > 2 soldiers shoot at it is 1- (1 — р)". (4.2) Evaluate the probability in (4.2) in the case when p = 0.0006 and n = 750, and then round your result to a…What is the definition of independence for two discrete random variables X and Y? X and Y are independent if and only if P(X= x) = P(Y= y) for all x and y. OX and Y are independent if and only if P(X = x Y = y) = P(X = x)*P(Y= y) for all x and y. X and Y are independent if and only if P(X= x | Y = y) = P(X = x) for all x and y. Previous
