Department of Mathematics
MTL 106/MAL 250 (Probability and Stochastic Processes)
Tutorial Sheet No. 3
1. X has a uniform distribution over the set of integers {−n, −(n − 1), . . . , −1, 0, 1, . . . , (n − 1), n}. Find the
distribution of (i) |X| (ii) X 2 (iii) 1/1 + |X|.
(
)2
2. If X has N (µ, σ 2 ), find the distribution of Y = a + bX, and Z = X−µ
.
σ
3. Let X be uniformly distributed random variable on the interval (0, 1). Define Y = a + (b − a)X, a < b.
Find the distribution of Y .
4. Let X be a random variable with pdf f (x) =
( )
of random variable Y = ln X
θ .
αθ α
(x+θ)α+1 ,
x > 0 where θ > 0 and α > 0. Find the distribution
(
)2
5. Let X be an random variable having an exponential distribution with parameter λ > 0. Let Y = X − λ1 .
Find the pdf of Y .
6. Let X be the life length of an electron tube and suppose that X may be represented as a continuous random
variable which is exponentially distributed with parameter λ. Let pj = P (j ≤ X < j + 1). Show that pj is
of the form (1 − α)αj and determine α.
7. Consider the marks of MAL 250 examination. Suppose that marks are distributed normally with mean 76
and standard deviation 15. 15% of the best students obtained A as grade and 10% of the worst students fail
in the course. (a) Find the minimum mark to obtain A as a grade. (b) Find the minimum mark to pass the
course.
8. Consider a nonlinear amplifier whose input X and output Y are related by its transfer characteristic
{
1
X2,
X>0
Y =
1
−|X| 2 , X < 0
Find pdf of Y if X has N (0, 1) distribution.
9. Let the phase X of a sine wave be uniformly distributed in the interval (− π2 , π2 ). Define Y = sin X. Find the
distribution of Y .
10. Let X be a random variable with uniform distribution in the interval (−π/2, π/2). Define
X ≤ −π/4
−1
tan(X) −π/4 < X < π/4
Z=
1
X ≥ π/4.
Find the distribution of the random variable Z.
11. Find the probability distribution of a binomial random variable X with parameter n, p, truncated to the
right at X = r, r > 0.
12. Find pdf of a doubly truncated normal N (µ, σ 2 ) random variable, truncated to the left at X = α and to the
right at X = β.
13. State True or False with valid reasons for the following statements.
(a) Let X be a discrete random variable with taking values
V ar(X) exists.
3k
,
2k
k = 0, 1, . . . and such that P (X =
(b) The MGF of a discrete random variable Y is given by MY (t) =
1 −3t
10 e
4t
+ 51 e−t +
2
5
+
3 2t
10 e
3k
)
2k
.
(c) If the characteristic function of a random variable W is φW (t) = e , then P (1 < W ≤ 5) =
1
1
4
.
=
1
.
2k+1
14. Prove that for any random variable X, E[X 2 ] ≥ [E[X]]2 . Discuss the nature of X when one have equality?
15. Suppose that two teams are plying a series of games, each of which is independently won by team A with
probability 0.5 and by team B with probability 0.5. The winner of the series is the first team to win four
games. Find the expected number of games that are played.
16. Let X be a random variable having a Poisson distribution with parameter λ. Prove that, for n = 1, 2, . . .
E[X n ] = λE[(X + 1)n−1 ].
17. A certain alloy is formed by combining the melted mixture of two metals. The resulting alloy contains a
certain percent of lead, say X, which may be considered as a random variable. Suppose that X has the
following pdf
{
6 ∗ 10−6 x(100 − x) 0 ≤ x ≤ 100
f (x) =
.
0
otherwise
Suppose that P , the net profit realised in selling this alloy per pound, is the following function of the percent
content is lead: P = C1 + C2 X. Compute the expected profit (per pound). Also find the variance of the
profit P .
18. Let X be a random variable having a binomial distribution with parameters n and p. Prove that
(
)
1
1 − (1 − p)n+1
E
.
=
X +1
(n + 1)p
19. Let X be a continuous random variable with CDF FX (x). Define Y = FX (X).
(a) Find the distribution of Y .
(b) Find the variance of Y , if it exist?
20. Suppose that X is a continuous random variable having the following pdf:
{ ex
x≤0
2 ,
f (x) =
e−x
,
x > 0.
2
Let Y = |X|. Obtain E(Y ) and V ar(Y ).
21. The mgf of a r.v. X is given by MX (t) = exp(µ(et − 1)).
(a) What is the distribution of X? (b) Find P (µ − 2σ < X < µ + 2σ), given µ = 4.
22. Let X be exponentially distributed random variable with parameter λ > 0.
(
)
(a) Find P | X − 1 |> 1 X > 1
(b) Explain whether there exists a random variable Y = g(X) such that the cumulative distribution function
of Y has uncountably many discontinuity points. Justify your answer.
23. Assume that, taxis are waiting in a queue for passengers to come. Passengers for these taxis arrive according
to a Poisson process with an average of 60 passengers per hour. A taxi departs as soon as two passengers
have been collected or 3 minutes have expired since the first passenger has got in the taxi. Suppose you get
in the taxi as first passenger. What is the distribution of your waiting time for the departure? Also, find its
variance.
24. The moment generating function of a discrete random variable X is given by MX (t) = 61 + 12 e−t + 13 et . If µ
is the mean and σ 2 is the variance of this random variable, find P (µ − σ < X < µ + σ).
25. Using MGF, find the limit of Binomial distribution with parameters n and p as n → ∞ such that np = λ so
that p → 0.
2
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