Let X and Y be independent, identically distributed exponential randomvariables with parameter A, and let Z=X+Y.a. Compute the conditional pdf of X and the conditional expected value of X giventhat Z=z.b. Compute the conditional pdf of X and the conditional expected value of X giventhat Z>t.

It is given that X and Y are independent and identically distributed exponential random variables…

Answered: Let X and Y be independent, identically…

A First Course in Probability (10th Edition)

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Author:Sheldon Ross

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Chapter1: Combinatorial Analysis

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Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...

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(a) For the hierarchical model YA Poisson (A) and 2 A gamma(a, B) 2 find the marginal distribution, mean, and variance of Y. Show that the marginal distribution of Y is a negative binomial if a is an integer. (b) Show that the three-stage model YIN binomial(N, p), NA~ Poisson(A), and leads to the same marginal (unconditional) distribution of Y. gamma(a, 3)
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a) Let X₁, X₂, X₁,...,X, be a random sample of size n from population X. Suppose that X-N(0, 1) and Y = -1-8√n. i) Show that the standard score of the sample mean X, is equal to Y. ii) Show that the mean and variance of the random variable Y are 0 and 1, respectively. iii) Show using the moment generating function technique that Y is a standard normal random variable. iv) What is the probability that Y2 is between 0.02 and 5.02?
3. Let the random variable X have the moment generating function M(t) = What are the mean and the variance of X, respectively? -1 < t < 1.
Assume that a random variable Y has a mean of zero and standard deviation of one. Calculate the expected value of X1 and X2 if they are defined as below. X1 = Sigma+ Mu X2 = Sigma(Y+Mu)
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Let X11 X12, Xini and X21, X22, X2n2 be two independent random samples of size n₁ and n₂ from two normal populations N(₁, 2) and N(2, 2) respectively. (a) Derive the maximum likelihood estimators (mle's) of all the parameters in the first population (X₁). Using analogy, state the mle's of the parameters of the second population. (b) Find the pooled estimator of the common variance when it is assumed that of = = ². Suggest an unbiased estimator of o². |
ans Theorem 9.3. Let X and Y be independent random variables with finite variances, and a, b ER. Then Var(aX) = a²Var (X), Var (X+Y) = VarX + Var Y, Var (ax + bY) = a²VarX + b²Var Y. sercise Prove the theorem. Remark 9.2. Independence is sufficient for the variance of the sum to be equal to the sum of the variances, but not necessary. Remark 9.3. Linearity should not hold, since variance is a quadratic quantity. Remark 9.4. Note, in particular, that Var (-X) = Var(X). This is as expected, since switching the sign should not alter the spread of the distribution.
(a) For the hierarchical model YA Poisson (A) and 2 A gamma(a, B) 2 find the marginal distribution, mean, and variance of Y. Show that the marginal distribution of Y is a negative binomial if a is an integer. (b) Show that the three-stage model YIN binomial(N, p), NA~ Poisson(A), and leads to the same marginal (unconditional) distribution of Y. gamma(a, 3)

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Let X and Y be independent, identically distributed exponential random variables with parameter A, and let Z-X+Y. a. Compute the conditional pdf of X and the conditional expected value of X given that Z- b. Compute the conditional pdf of X and the conditional expected value of X given that Z>L