Let M (t) be the moment generating function of the random variable X with probability mass function p (1) = , p (2) = }, and p (3) =. Then M (1) is equal to
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- Let X~ exp(X) be an exponential random variable. (a) Compute P(X > t) for some positive real number t. (b) Compute P(X > t+s|X > s) for some positive real numbers t and s. (c) If for a random variable X, P(X > t) = P(X > t + s|X > s), we call it memoryless. Justify this name. Is exponential distribution a memoryless random variable?Establish the following: 0P (E)1 , P (A u B) = P (A) + P (B) – P (A n B) , P (A) = 1 – P (A’) ii The probability that a boy with a catapult hits target A is 2/3and that he hits target B is ¾. Given the probability of hitting both targets to be 1/2, find the probability that he (a) hits at least one of the targets (b) does not hit any. iii What is the probability of having at least one 6 in three throws with a dice?Let Y be a Poisson random variable with mean λ = 2. (a) Find P(Y ≥ 2). (b) Find P(Y ≥ 4|Y ≥ 2).
- (1) Let X = b(16, -) find E(4-3x) and distribution function. (2) Let X be a random variable with p.d.f. -2 kx 1A continuous random variable X has a P.D.F. fx (x) = 3 X, 0Sr a) and (ii) P (x > b) = 0.05.Consider the random variable X with PDF (known as Cauchy distri- bution) f(x) = 7 - 003 Let X be a random variable with probability Law P (X = r) = q, K for r = 1,2,3.. . 00, Find the moment generating function and mean of ihe random variable X. Let K+ q = 1.Suppose X and Y are independent random variables with E(X) =2, E(Y)=3,V(X)=4,V(Y)=16. Finda)E(5X-Y) b)V(5X-Y) c)COV(3X+Y,Y) d)COV(X,5X-Y)Suppose that Z1, Z2, . . . , Zn are statistically independent random variables. Define Y as the sum of squares of these random variablesRecommended textbooks for youA First Course in Probability (10th Edition)ProbabilityISBN:9780134753119Author:Sheldon RossPublisher:PEARSON
A First Course in Probability (10th Edition)ProbabilityISBN:9780134753119Author:Sheldon RossPublisher:PEARSON