Let X and Y be two independent random variables such that X~Gamma (m, 2) and Y~Gamma...

 

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Let X and Y be two independent random variables such that X~Gamma (m, 2) and Y~Gamma (n 1). Prove that if W = X + Y, then W~Gamma (m + n, 2) through the convolution of X and Y, that is, find the expression for the density function fw(w). 1. Hint: fx+y = fx(w-y). fy(y) dy This is the expression of the convolution.. Then use the indicated change of variable and remember that B(a, b) = Where the function Beta is B(a, b) = xa-1. (1-x)b-1 dx 2. Let X be a random variable X~F(a, b) and Y= =, show that Y also follows a distribution F of Fisher,Y~F(b, a). r(a)r (b) r(a+b) Let X and Y be two independent random variables such that X~Gamma (m, 2) and Y~Gamma (n 1). Prove that if W = X + Y, then W~Gamma (m + n, 2) through the convolution of X and Y, that is, find the expression for the density function fw(w). 1. Hint: fx+y = fx(w-y). fy(y) dy This is the expression of the convolution.. Then use the indicated change of variable and remember that B(a, b) = Where the function Beta is B(a, b) = xa-1. (1-x)b-1 dx 2. Let X be a random variable X~F(a, b) and Y= =, show that Y also follows a distribution F of Fisher,Y~F(b, a). r(a)r (b) r(a+b)