A random variable X depending on parameters v > 0,1> 0 has a probability density function that can be expressed as an in

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A Random Variable X Depending On Parameters V 0 1 0 Has A Probability Density Function That Can Be Expressed As An In 1 (42.69 KiB) Viewed 22 times

A random variable X depending on parameters v > 0,1> 0 has a probability density function that can be expressed as an infinite series () =Σ fx (2) = (1/2)* 2-1/2 k! - 9v+2(2) k=0 where ga(2),d > 0 is the probability density function of a xả random variable. Let mx (t) denote the moment generating function of X and ra(t) denote the moment generating function of a xả random variable. (i) Show that mx(t) k! -Pv+2*(t) 00 (1/2)*e-1/2 k=0 [4 marks] (ii) Hence, or otherwise, show that mx(t) 1 (1 - 2t)/2 exp Xt 1-2t [8 marks] (iii) Find E{X} in terms of v, A. [6 marks] (iv) If X1,..., 1,..., X, are iid random variables with density fx (2), how is - X, distributed? [7 marks]