Suppose X Gamma (a; b) and YGamma (c; d). Let W-X+Y. (a) Find the MGF of...
Suppose X ∼ Gamma(a, b) and Y ∼ Gamma(c, d). Let W = X + Y . (a) Find the MGF of W. (b) What restrictions would need to be placed on the values of a, b, c, and d in order for W to be a Gamma Random Variable. What would the parameters be?
uppose XGamma(a, b) and Y Gamma(c,d). Let W -X +Y. (a) Find the MGF of W. b) What restrictions would need to be placed on the values of a, b, c, and d in order for W to be a Gamma Random Variable. What would the parameters be?
1. Suppose X ∼ Gamma(a,b) and Y ∼ Gamma(c,d). Furthermore suppose X and Y are independent. Let W = X + Y . (a) Find the MGF of W. (b) What restrictions would need to be placed on the values of a, b, c, and d in order for W to be a Gamma Random Variable. What would the parameters be?
In details plz Thank you! Suppose that X ~ Gamma(a, b) and Y ~ Chisquare(k) and X and Y are independent. Îet w = X+Y (a) Find the MGF of W. / (b) For what value(s) of b would W be a Gamma Random Variable? What would its parameters be? (c) For what value(s) of b and a would W be a ChiSquare Random Variable? What would its parameter be? (d) For what value(a) of b, a, and k would...
The moment generating function (MGF) for a random variable X is: Mx (t) = E[e'X]. Onc useful property of moment generating functions is that they make it relatively casy to compute weighted sums of independent random variables: Z=aX+BY M26) - Mx(at)My (Bt). (A) Derive the MGF for a Poisson random variable X with parameter 1. (B) Let X be a Poisson random variable with parameter 1, as above, and let y be a Poisson random variable with parameter y. X...
Having troubles with question 2. Please help 2. If X has a Gamma distribution with parameters a and B, then its mgf is given by (a) Obtain expressions for the moment-genérating functions of an exponential random variable and of a chi-square random variable by recognizing that these are special cases of a Gamma distribution and using the mgf given above. (b) Suppose that X1 is a Gamma variable with parameters α1 and β, X2 is a Gamma variable with parameters...
Suppose that a rv Y has mgf m(t)- (a) 1-bt) Differentiate this mgf twice and thereby obtain the mean and variance of Y. [5 marksj] (b) Suppose m(t) is the mgf of a rv W. Let r(t) be the natural logarithm of m(t), ie·r(t) = login(1). Find r'() and r"(t), and express r'(0) and r"(0) in terms of EW and VarW. [5 marks] Use the result in (b) to find the mean (d) Find the mean and variance of the...
7. The Gamma distribution is commonly used to model continuous data. The probability density function of a Gamma random variable is f (zlo, β)- a. Find the MGF of a Gamma random variable. b. Use the MGF to find the mean of a Gamma random variable. c. Use the MGF to find the second raw moment of a Gamma random variable. d. Use results (b) and (c) to find the variance of a Gamma random variable. e. Let Xi, í...
(1 point) In Unit 3, I claimed that the sum of independent, identically distributed exponential random variables is a gamma random variable. Now that we know about moment generating functions, we can prove it. Let X be exponential with mean A 4. The density is 4 a) Find the moment generating function of X, and evaluate at t 3.9 The mgf of a gamma is more tedious to find, so l'll give it to you here. Let W Gamma(n, A...
1. Suppose that Y ∼ Gamma(α, β) and c > 0 is a constant. (a) Derive the density function of U = cY. (b) Identify the distribution of U as a standard distribution. Be sure to identify any parameter values. (c) Can you find the distribution of U using MGF method also? I. Suppose that Y ~ Gamma(α, β) and c > 0 is a constant. (a) Derive the density function of U cY. (b) Identify the distribution of U...