F71SM STATISTICAL METHODS
Tutorial on Section 3 RANDOM VARIABLES
1. A discrete random variable X has probability mass function x012
f (x) 0.25 0.5 0.25
Find the mean, variance, and probability generating function of X.
[E[X] = 1, Var[X] = 0.5, Pgf: G(t) = (1 + t)2/4]
2. A discrete random variable X has probability mass function
−2 2x
f(x) = e x! x=0,1,2,…
Find the mean, variance, probability generating function, and moment generating function of X. [E[X] = 2, Var[X] = 2, Pgf: G(t) = e2(t−1), Mgf: M(t) = exp(2(et − 1))]
3. A series of independent trials, each with probability p of ‘success’, is continued until the second success is obtained. Let X be the number of trials required.
Find the probability generating function, and the mean and standard deviation of X. √
Letq=1−p,G(t)=p2t2(1−qt)−2 for −1
1
(a) Show that the moment generating function of X is given by M(t) = (1 − t2)−1, for
−1 < t < 1, and hence find the mean and standard deviation of X by (i) expanding
M(t) as a power series in t, and (ii) by differentiating M(t) and putting t = 0.
√
2]
(b) What is the mgf of the r.v. Y , where Y = 2X +3? What are the mean and standard
[E[X] = 0, Var[X] = 2, SD[X] = deviation of Y? [E[Y] = 3, SD[Y] = 2 2]
8. Let X be a continuous r.v. and let Y = X2.
By considering the cumulative distribution function of Y , FY (y) = P (Y ≤ y) = P (X2 ≤
y), show that the pdfs of Y and X are related by
fY(y) = 21y−1/2(fX(√y)+fX(−√y)), y>0
HencefindthepdfofY =X2 whereX haspdf 1 −x2/2
√
f(x)=√e ,−∞