F71SM STATISTICAL METHODS
Tutorial on Section 3 RANDOM VARIABLES
1. A discrete random variable X has probability mass function
x 0 1 2
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
f(x) = e−2
2x
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.[
Let q = 1− p,G(t) = p2t2(1− qt)−2 for − 1 < qt < 1, µ = 2/p, σ =
√
2(1−p)
p
]
4. See tutorial on section 2, Q9. Given that player A wins, find the probability that he does
so on his rth throw, and hence show that the expected number of times player A throws
in a game which he wins is approximately 5.8.
5. An investor’s income in a year from his investments (X, in units of £1,000) is a random
variable with probability mass function
x 16 18 20 22
f(x) 0.1 0.2 0.5 0.2
He pays tax on his returns at 25% on any income in excess of £3000.
Find the probability distribution of his net income (income after tax), and calculate the
means of his gross and net incomes. [E[X] = 19.6(£19, 600), E[Y ] = 15.45(£15, 450)]
6. Consider the random variable with probability density function
f(x) =
c
x3
, 1 < x < 2 (= 0 otherwise).
(a) Show that c = 8/3, and find E[X], E[X2], Var[X], and SD[X].
[E(X) = 4/3, E[X2] = (8/3) ln 2,Var[X] ≈ 0.0706, SD[X] ≈ 0.266]
(b) Find the distribution function F (x).
7. Let X be a random variable with pdf
f(x) =
{
ex/2 x ≤ 0
e−x/2 x > 0
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.
[E[X] = 0, Var[X] = 2, SD[X] =
√
2]
(b) What is the mgf of the r.v. Y , where Y = 2X+3? What are the mean and standard
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) =
1
2
y−1/2 (fX(
√
y) + fX(−
√
y)) , y > 0
Hence find the pdf of Y = X2 where X has pdf
f(x) =
1
√
2π
e−x
2/2, −∞ < x <∞. 2