4.Let X and Y be independent random variables with gamma distributions,r(3,2) and I'(4,2), respectively. The moment generating function of X andY are• (1-2₁)a)b)c)d)and¹ (1-2)* · · .

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A First Course in Probability (10th Edition)

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Author:Sheldon Ross

Publisher:Sheldon Ross

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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Let X1, X2, ... be independent Cauchy random variables, each with PDF d f(x)= TT(d² +x?)' Show that A,= (X, +X2+•…+X„)/n has the same distribution as X,.
Let X1, X2,.., X, be independent identically distributed random variables with each X, having a probability mas function given byP(X; = 0) = 1-p P(X; = 1) = p, where Osps1. 1 EXj, then E(Y)= j = 1 Define the random variable Y = n Select one: а. 1 b. p C. 0 d.
2. Suppose that X1, . Xn are iid Geometric random variables with frequency func- ... tion f(x; 0) = 0(1 – 0)", x = 0, 1, 2, ..., 0 E (0, 1). Find the ML estimator 0, of 0. Show that Ôn is consistent an find its asymptotic distribution.
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?
Suppose X1, X2, ... , Xn is a random sample and Xi = {1, with probability p 0, with probability 1-p} for every i = 1, 2, ... , n. Find the Moment Generating Function of ∑i=1n Xi . What is the distribution of ∑i=1n Xi ?
3. Let X1,.., Xn be a random sample from a uniform distribution on the interval [a, b). In other words, X U(a, b). Obtain method of moments estimators for a and b.
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Q3. Generate three random variates according to each of the following distributions using your own uniform random variates from U(0, 1) and list them: • Discrete Uniform: U (15, 17) • Uniform: U (0.57, 1.5). • Triangular: Triang (2, 10, 4). • Binomial: Bin (12, 0.4). • Lognormal: LN (5, 1). Beta: Beta (5, 5). • Negbin (8, 0.4) • Weibull (.5, 2.5) Geom (0.5) • PT5 (0.5, 0.5)
5. Let X₁ and X2 be independent random variables. Suppose that X₁ follows a Poisson distribution with parameter X₁, and that X2 follows a Poisson distribution with parameter λ₂. Let Y = X₁ + X₂. (a) Use the MGF technique to find the distribution of Y. (b) Find E(Y) and Var(Y).
Suppose Y is a continuous random variable drawn from the uniform distributionon the interval [3, 4], that is, Y ∼ Uniform([3, 4]). Conditioned on Y = y, a second randomvariable X is drawn from the uniform distribution on the interval [0, y]. What is fX(x), thepdf of X?

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4. Let X and Y be independent random variables with gamma distributions, r(3,2) and I'(4,2), respectively. The moment generating function of X and Y are *(1-2)' (1) and ().<. respectively. a) b) c) Calculate E(3X + 2Y+1). Calculate Var (3X + 2Y+1). Obtain the distribution of W = X+Y. (1)