2. For the following discrete variable with probability f(x) X 2 3 4 f(x) 0.2 0.15 2c 0.05 0.1 (a) Determine the value of c
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- 1. Suppose that the amount X dispensed by a beverage-dispensing machine has a uniform probability distribution on [a, b] (in Ounces). (a) Given a and b, find x0 such that P(X < μ+x0) = 0.90, where μ = E[X]. (b)Given an i.i.d. sample X1, . . . , X200, explain how to estimate θ = b − a. Is the proposed estimator unbiased?If a probability generating function of a random variable x is Px(s)=(1/3+2/3s)^6 determine E(x),var(2x) and pr(X>1)Translate the Binomial Probability to Normal Probability (Discrete to Continuous) a. Less Than: P(x<175) is equivalent to P(x < )b. Greater Than: P(x>157) is equivalent to P(x > )c. At Least: P(x≥89)is equivalent to P(x > )d. At Most: P(x≤160) is equivalent to P(x < )e. Between: P(92<x<295) is equivalent to P( < x < )
- Let X be a Poisson random variable with E(X) = 3. Find P(2 < x < 4).Suppose that X is an exponential random variable with mean 5. (The cumulative distribution function is F(x) = 1- e-x/5 for x >= 0, and F(x) = 0 for x < 0. (a) Compute P(X > 5). (b) Compute P(1.4 <= X <= 4.2). (c) Compute P(1.4 < X < 4.2).When a certain glaze is applied to a ceramic surface, the probability is 5% that there will be discoloration, 20% that there will be a crack, and 23% that there will be either discoloration or a crack, or both. Let X = 1 if there is discoloration, and let X = 0 otherwise. Let Y = 1 if there is a crack, and let Y = 0 otherwise. Let Z = 1 if there is either discoloration or a crack, or both, and let Z = 0 otherwise. a) Let pX denote the success probability for X. Find pX. b) Let pY denote the success probability for Y. Find pY. c) Let pZ denote the success probability for Z. Find pZ. d) Is it possible for both X and Y to equal 1? e) Does pZ = pX + pY? f) Does Z = X + Y? Explain.
- Suppose X has probability distributionx: 0 1 2 3 4P(X = x) 0.2 0.1 0.2 0.2 0.3Find the following probabilities:d. P(X = 1 or X ≤ 3)e. P(X = 2 given X ≤ 2)If X has the Poisson distribution with P(X=1) = 2P(X=2), then P(X ≥ 2) is approximately:Suppose X and Y are jointly discrete randam variables, the conditional expectation of v(y),given that X= x,is