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- ./Let (X1, X2, X3) be the random outcome of tossing a fair six-faced die three times, and let (Y(1), Y(2), Y(3) be the order statistics from (X1, X2, X3). We have Y(1) = min (X1, X2, X3), Y(3) = max (X1, X2, X3), and Y(1) < Y(2) < Y(3). Find Pr (Y(1) = 3).Let X1, X2,..., Xn be independent exponential variables, parameter λ. Show by induction that X2 S = X₁ + X₂ + + Xn has the r(λ, n) distribution.
- Suppose that X and Y are random variables with E(X) = 1, E(X²) = 5, E(Y) = 3, and E(XY) = 1. Match up the following quantities. Cov(X,Y) Corr(X, Y) Var(-X/2) Cov(X/2,-2Y + 3) Drag answer here Cannot be determined Drag answer here Drag answer here 2 Cannot be determined 1COMPUTE RANGE * From the 50 scores below, 48 56 53 60 43 65 62 72 38 66 32 37 60 49 56 62 73 63 52 52 38 42 56 71 48 59 42 55 66 63 70 35 39 67 32 48 37 38 39 73 63 50 48 50 44 50 52 43 42 70 50 41 32 7319
- 2. Suppose one has n non-degenerate random variables X1, X2,, X, so that X1+ X2 + · · · + Xn = L for some constant L. (Recall that a random variable is non-degenerate if it is not a constant in disguise.) Show that there must be at least one pair of indices i j so that p(X;, X;) < 0.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.How many ways can 6 joggers line up single-file to get a drink from a water fountain?
- 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…7. Suppose that the joint p.d.f. of two random variables X and Y is as follows: (4 – 2x – y) for x > 0, y > 0, f(x, y) = and 2x + y 2|X = 0.5).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