(a) Let X be a random variable with PDF ƒ(x) = 1−|x|, |x| ≤ 1. Obtain the MGF of X. (b) Find the PDF of a random variable Y with MGF T 2 My(t) = 1 = ( ³² = ¹) ²³₁ t

If X is an exponential random variables with rate 1, then its distribution function is given by                                F(x) = 1 − e−x Show that                                x = − ln(1 − u).

A random process X(t) is applied as input to a system whose impulse response is h(t) 3u(t)t² exp (-8t). If E[X(t)] = 2, what is the mean value of the system response y(t). - CS Scanned with CamScanner

(b) Let X1, X2, ...,X, be a random sample from the pdf Be z for z 2 0 f(z,0) = 0, otherwise (1) Show that the statistic T(X1, X2, . . , Xn) = X, is complete sufficient for 8. %3D i=1

(c) Let X follow the gamma distribution with pdf S(z) = r(a) on z> 0. If a =1 and V(X)=4 find the value of b, and hence find E(X).

The Exponential pdf for continuous random variable Y takes the form e-ylß y > 0 f(V) = for ß > 0. y 1.6) Give your answer to three decimal places.

(b) The probability density of a function is given by p(x) from x = 1 to x = 4. That is 62 x3 dx = 1 62 P(x) dx = - 00 Determine the value of the standard deviation.

x > 1 A RV X has the PDF fx(x) = otherwise (a) Find the MGF Mx(t) of X and clearly define the values of t for which it exists. (b) Find the mean (µ) and variance (o?) of the RV X using the MGF from (a).

Solve both parts

1. Fix r € N, and suppose that Y₁,..., Yn are independent and identically distributed NegBin(r, ) random variables, each with probability mass function given by ¡v (u | 0) – (1 + r − ¹) (1 - fy y (1 - 0) ³0⁰ for y ¤ N and unknown 0 € (0, 1). Here the number of successes r in the negative binomial is known.