In the name of GOD.
Sharif University of Technology
Stochastic Processes Dr. H.R. Rabiee
CE 695 Fall 2013
Homework 1 (Review Problems)
1. If A1 , A2 , · · · , Ak are events with P (A1 ∩ A2 ∩ · · · ∩ Ak−1 ) > 0, prove that:
P (∩ki=1 Ai ) = P (A1 )P (A2 |A1 )P (A3 |A1 ∩A2 ) · · · P (Ak |A1 ∩A2 ∩· · ·∩Ak−1 ).
2. Assume that A1 , A2 , A3 , A4 are independent events and P (A3 ∩ A4 ) > 0,
show that:
P (A1 ∪ A2 |A3 ∩ A4 ) = P (A1 ∪ A2 )
3. Verify that a function defined by:
(
a × 34 ( 14 )x
P (x) =
0
x = 0, 1, 2, ...
o.w.
Is a PMF of a discrete r.v X and find:
• P (X = 2)
• P (X ≤ 2)
• P (X ≥ 1)
4. Suppose that X is a random variable with probability density function

2

1≤x<2
3x − a
fX (x) = − 31 x + b 2 ≤ x ≤ 4


0
o.w.
and assume that E[X] = 37 .
(a) Find the values of a and b which make fX (x) a valid probability
density function.
(b) Find FX (x) and show that FX (x) has the properties of a cumulative
distribution function.
(c) Find P(X = 3).
(d) Find P( 1.5 ≤ X < 4).
(e) Find P(1.5 ≤ X < 4|X ≥ 3).
5. Show that Cov(aX + b, cY + d) = ac Cov(X, Y ).
1
6. Suppose X and Y have joint density
(
1 −x/y −y
e
e
fXY (x, y) = y
0
x > 0, y > 0
o.w.
Find P (X > 1|Y = y).
7. Suppose that X and Y are two random variables.
(a) Prove that if X and Y are independent, they are also uncorrelated.
(b) Give an example which X and Y are uncorrelated but not independent.
(c) If X and Y are normally distributed and independent with the same
variance, prove that X − Y and X + Y are independent.
8. Let X represent a uniform random variable. Find fY (y) if
(a) X ∼ U (−2π, 2π) and Y = X 3 .
(b) X ∼ U (−2π, 2π) and Y = X 4 .
(c) X ∼ U (−π/2, π/2) and Y = tan(X).
9. Express the density fY (y) of the random variable Y = g(X) in terms of
fX (x) if
(a) g(x) = |x|.
(b) g(x) = e−x U (x).
10. Assume that X and Y are two independent exponential random variables
with parameters λ and µ respectively.
(a) Find the probability density function of Z = max(X, Y ).
(b) Find the probability density function of Z = min(X, Y ).
11. Suppose that X and Y are two independent random variables.
√ If X ∼
2Y cos (X)
unif orm(0,
2π)
and
Y
∼
exponential(1),
Show
that
W
=
√
and Z = 2Y sin (X) are independent random variables with standard
normal distribution.
12. Consider two independent continuous random variables X and Y with
PDFs fX and fY , respectively and let Z = X + Y .
(a) Show that fZ|X (z|x) = fY (z − x).
(b) Assume that X and Y are exponentially distributed with parameter
λ, find fX|Z (x|z).
13. For any two random variables X and Y , prove that:
(a) E[E[X|Y ]] = E[X].
(b) var(X) = E[var(X|Y )] + var(E[X|Y ]).
14. Consider two random variables X and Y with positive variances.
2
(a) prove that |ρ(X, Y )| ≤ 1 where ρ(X, Y ) is the correlation coefficient
between X and Y .
(b) Show that if Y − E[Y ] is a positive or negative multiple of X − E[X],
then ρ(X, Y ) = 1 or ρ(X, Y ) = −1 respectively.
15. Assume that X1 , X2 , ... are identically distributed random variables with
expectation µ and variance σ 2 . Let N be a positive integer-valued random
variable and Y = X1 + ... + XN .
(a) Show that E[Y ] = µE[N ].
(b) Show that var(Y ) = σ 2 E[N ] + µ2 var(N ).
(The random variables X1 , X2 , ... and N are all independent.)
16. Papoulis: 5-27,5-35.
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