FUNCTIONS OF RANDOM VARIABLE
1. Find the distribution function of the RV Y = g(X), in terms of
distribution function of X. if it is given that
x − c for x > c
g(x) = { 0 for|x| ≤ c}
x+c x < −c
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2. The random variable Y is defined by Y = (X + |X|), where X is
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another RV. Determine the density and distribution function of Y in
terms of those of X.
3. a) Find the density function Y =aX + b in terms of the density
function of X
b) Let X be a continuous RV with pdf
x
f(x) = 12 , in 1 < x < 5
= 0, elsewhere
find the probability density function of Y = 2X – 3.
4. If X is a continuous RV with some distribution defined over (0, 1)
such that P (X ≤ 0.29) = 0.75, determine k so that
P (Y ≤ k) = 0.25 where Y = 1 – X.
5. If Y=X², where X is a Gaussian random variable with zero mean and
variance σ2, find the pdf of the random variable Y.
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6. If the continuous RV X has a pdf fx(x)=9 (x+1) in -1 < x < 2 and = 0,
elsewhere, find the pdf of Y = X2.
7. Given the RV X with density function
2x, 0<x<1
F(x) = {
0, elsewhere
find the pdf of Y = 8X3.
8. If X is a Gaussian random variable with mean zero and variance σ2,
find the pdf of Y = |X|.
F y(y) = P (Y ≤ y) = P (|X| ≤ y)
= P {-y ≤ X ≤ y}
= F x(y)-F x(-y)
9. (i) If X is a normal RV with mean zero and variance σ2, find pdf of
Y = eX.
(ii) If X has an exponential distribution with parameter α, find the
pdf of Y = log X.
10. If X and Y are independent RVs having density functions,
2e−2x , x≥0
f1(x) = { and
0 , x<0
3e−3y , y≥0
f2(x) = { and
0 , y<0
find the density function of their sum U= X+Y
11. If fx(x)=ce-cxU(x) and fz(z)=c2ze-czU(z), find fy(y),if Z= X + Y and X and Y are independent.
12. If X and Y are independent RVs and if Y is uniformly distributed in
(0, 1), show that the density of Z = X + Y is given by
fz(z) = Fx(z) – Fx(z-1).
13. The current I and the resistance R in a circuit are independent
continuous RVs with the following density functions.
fi(i) = 2i, 0≤i≤1
=0 elsewhere
fr(r) = r2/9 0≤r≤3
=0 elsewhere
Find the pdf of the voltage E in the circuit, where E = IR.
14. If X and Y are independent RVs each following N(0, 2) find the pdf
of Z = 2X +3Y.
15. If X and Y each follow an exponential distribution with parameter
1 and are independent, find the pdf of U = X – Y.
Fx(x) = e -x, x > 0, and fy (y) = e-y, y > 0
16. If the joint pdf of (X, Y) is given by fxy(x, y) = x + y; 0 ≤ x, y≤ 1, find
the pdf of U=XY.
17. If X and Y are independent RVs with fx(x) = e-x U(x) and fy(y) =3e-3y
X
U(y), find fz(z), If Z = Y .
18. If X and Y are independent RVs with pdf’s e−x ,x ≥ 0, and e−y , y ≥ 0
X
respectively find the density functions of U = and V = X + Y.
X+Y
Are U and V independent?
x 2 ⁄2α2
19. If the continuous RV X has density f(x)=α2 e−x ×U(x),
n
find E(X ) and deduce the values of E(X) and var(X).
20. A line of length a units is divided into two parts. If the first part is
length X, find E(X), var(X) and E{X(α - X)}.
21. If X is a continuous RV, prove that
∞ 0
E(X) = ∫0 [1 − F (x)]dx -∫−∞[1 − F(x) ]dx
22. If the random variable X follows N(0, 2) and Y = 3X2, find the
mean and variance of Y.
23. Two cards are drawn at random with replacement from a box
which contains 4 cards numbered 1, 1, 2 and 2. If X denotes the
sum of the numbers shown on the two cards, find the mean and
variance of X.
24. If X and Y are two independent RVs with fx(x)=e-xU(x) and
fy(y) = e-y U(y) and Z = (X - Y) U(X - Y), prove that E(Z) = 1⁄2.
25. The joint pdf of (X, Y) is given by f(x, y)=24xy: x>0,y>0,
x+y≤1, and f(x,y)=0, elsewhere, find the conditional mean
and variance of Y, given X.
26. If (X, Y) is uniformly distributed over the semicircle bounded by
y = √1 − x 2 and y = 0, find E(X / Y) and E(Y|X), Also verify the
E[E(X/Y)] = E(X) and E[E(Y/X)] = E(Y).
27. If X represents the outcome, when a fair die is tossed, find the MGF
of X and hence find E(X) and Var(X).
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28. If a RV X has the MGF M(t)=3−t , obtain the standard derivation of X.
29. Find the MGF of the binomial distribution and hence find mean and
variance.
30. Find the density function f(x) corresponding to the characteristic
function defined as
1 − |t | for |t| ≤ 1
Φ(t) = {
0 for |t| > 1
31. The lifetime of a certain brand of an electric bulb may be considered
a RV with mean 1200 h and standard deviation 250 h. Find the
probability, using central limit theorem, that the average lifetime of
60 bulbs exceed 12500 h.
32. A distribution with unknown mean μ has variance equal to 1.5 Use
central limit theorem to find how large a sample should be taken
from the distribution in order that the probability will be at least
0.95 that the sample mean will be within 0.5 of the population mean.
