4. Suppose random variables X; ~ N(i, o?), for i = 1,... ,Let Y =L-1n, and X1, ... , Xn are mutually independent.(X;-µ;)²i-D1%3Dpg. 1 of 2(1) Find the PDF of Y.(2) Find the moment generating function of Y.

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Answered: 4. Suppose random variables X; ~ N(Hi,…

MATLAB: An Introduction with Applications

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Chapter1: Starting With Matlab

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2.2 Use the uniqueness of moment-generating funetions to give the distribution of a random variable with moment-generating function m(t) (0.8e' + 0.2)7.
2. Suppose that Y1 ~ Exp(3) and Y2 ~ Exp(2) are independent. (a) Determine the joint pdf of Y1 and Y2. (b) Use the Jacobian method to determine the pdf of U = Y1 + Y2. (c) Use the mgf method to determine the distribution of V = 4Y1 + 6Y2. The distribution of V will be one of the "famous" ones in your Super Handout.
10 If 0₁ <0₂, derive the moment-generating function of a random variable that has a uniform distribution on the interval (0₁, 0₂). Suppose U has a uniform distribution on the interval (0₁, 0₂). Then, the moment generating function of U is derived as follows. mu(t)= E = = ·0₂ 7⁰2 du
6. Suppose the moment generating function of X is M(t) = ² et + e²t+e³t. a) Find the mean of X. b) Find the variance of X. c) Find the pmf of X.
Suppose claim amounts at a health insurance company are independent of one another. In the first year calim amounts are modeled by a gamma random variable X with alpha=40, and beta=3. In the second year, individual claim amounts are modeled by random variable Y=1.05X+3. Let W be the average of 30 claim amounts in year two set up the equation to model the random variable W. a) Find the moment generating function of W b) Based on moment generating function of W is W also a gamma distribution? if so what are the parameters? c) Find the approximate probability that W is between 125$ and 130$.
69. Let X~ Geom (p) with pmf f(x) = P(X = x) = p(1 − p)ª−¹I{1,2,...} (x). (a) Use the definition of the moment generating function of a discrete random variable to find the moment generating function mx (t) for X. (b) Use the mgf for X to derive the formula for the mean of X, μ = E(X) = m'x (0).
11. Let (y1, Y2 ... Yn) be independent random sample from the uniform distribution on [0, 1]. (a) show that Z = – In Y; has exponential distribution with parameter 1. (b) Hence or otherwise, show that -2 In Y; xản
4.
1. If X,X2, .,X, forms a random sample of size n from a population with pdf f(x; 0,8) = for x>0 elsewhere Find estimators for 8 and 0 by the method of moments.
7. Suppose that X is a continuous random variable with pdf Se-(x+2), 2), if-2 < x < 0 f(x)= = lo, elsewhere a) Find the moment generating function of X. b) Use the MGF of (a) to find E(X) and E(X²). 8.
Suppose there are three possible states of nature, and the class-conditional PDFS are the Cauchy distributions 1 p(rw;) = 1 i = 1,2,3. (1) 2 1+ bị Let ai = -2, a2 = 0, az = 2, bị = b2 = 1, b3 = 0.5. Let P(wi) = P(w2) = 0.4, P(w3) = 0.2. (a) Find the decision boundaries and the decision regions for the Bayesian decision (b) Calculate the probability of error for the classification done according to this
Q. 3 Let continuous random variables X and Y be independent identically distributed random variables with the following respective pdfs: fx(x) = 2e-2x;x 2 0 and fy (y) = 2e-2y;y > 0 Define a new random variable Z = X + Y. Derive moment generating function of Z, i.e. Mz(t).

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4. Suppose random variables X; ~ N(Hi, o?), for i = 1,...,n, and X1,..., Xn are mutually independent. Let Y = (X,-)² pg. 1 of 2 47 (1) Find the PDF of Y. (2) Find the moment generating function of Y.