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MATH 105 921 Solutions to Probability Exercises
MATH 105 921 Solutions to Probability Exercises
Problem 1. Suppose we flip a fair coin once and observe either T for ”tails” or H for”heads”. Let X1 denote the random variable that equals 0 when we observe tails and equals1 when we observe heads. (This is called a Bernoulli random variable.)
(a) Make a table of the PDF of X1 and calculate E(X1) and Var(X1).
Solution: We have the following PDF table of X1:
x Pr(X1 = x)0 1
2
1 12
The expected value is:
E(X1) =
(1
2
)0 +
(1
2
)1 =
1
2.
The variance is:
Var(X1) =
(1
2
)(0− 1
2
)2
+
(1
2
)(1− 1
2
)2
=1
4.
(b) Now suppose that we flip a fair coin twice. We will observe one of four events: TT, TH,HT, or HH. Let X2 be the random variable that counts the number of heads we observe.(So, X2 can take the values 0, 1 or 2.) Redo part (a) for this new random variable.
Solution: We have the following PDF table of X2:
x Pr(X2 = x)0 1
4
1 12
2 14
The expected value is:
E(X2) =
(1
4
)0 +
(1
2
)1 +
(1
4
)2 = 1.
The variance is:
Var(X2) =
(1
4
)(0− 1)2 +
(1
2
)(1− 1)2 +
(1
4
)(2− 1)2 =
1
2.
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MATH 105 921 Solutions to Probability Exercises
(c) Let X3 be the random variable that counts the number of heads we observe after threesuccessive flips of a fair coin. Redo part (a) for this new random variable.
Solution: We have the following PDF table of X3:
x Pr(X3 = x)0 1
8
1 38
2 38
3 18
The expected value is:
E(X3) =
(1
8
)0 +
(3
8
)1 +
(3
8
)2 +
(1
8
)3 =
3
2.
The variance is:
Var(X3) =
(1
8
)(0− 3
2
)2
+
(3
8
)(1− 3
2
)2
+
+
(3
8
)(2− 3
2
)2
+
(1
8
)(3− 3
2
)2
=3
4.
(d) Do you notice any patterns in your calculations? Make a guess for the pdf table, theexpectation and the variance of X4, the random variable that counts the number ofheads we observe after four successive flips of a fair coin, and then verify your guess bydirect computation.
Solution: We have the following PDF table of X4:
x Pr(X4 = x)0 1
16
1 14
2 38
3 14
4 116
The expected value is:
E(X4) =
(1
16
)0 +
(1
4
)1 +
(3
8
)2 +
(1
4
)3 +
(1
16
)4 = 2.
Page 2 of 10
MATH 105 921 Solutions to Probability Exercises
The variance is:
Var(X4) =
(1
16
)(0− 2)2 +
(1
4
)(1− 2)2 +
+
(3
8
)(2− 2)2 +
(1
4
)(3− 2)2 +
(1
16
)(4− 2)2
= 1.
Problem 5. Recall the example of rolling a six-sided die. This is an example of a discreteuniform random variable, so named because the probability of observing each distinct out-come is the same, or uniform, for all outcomes. Let Y be the discrete uniform randomvariable that equals the face-value after a roll of an eight-sided die. (The die has eight faces,each with number 1 through 8.) Calculate E(Y ), Var(Y ), and StdDev(Y ).
Solution: We have the following PDF table of Y :
x Pr(Y = x)1 1
8
2 18
3 18
4 18
5 18
6 18
7 18
8 18
The expected value is:
E(Y ) =8∑
x=1
xPr(Y = x) =1
8
(8∑
x=1
x
)=
1
8
(8(9)
2
)=
9
2.
The variance is:
Var(Y ) =8∑
x=1
(x− E(Y ))2 Pr(Y = x) =1
8
(8∑
x=1
(x− 9
2
)2)
=1
8(42) =
21
4.
The standard deviation is:
StdDev(Y ) =√
Var(Y ) =
√21
4=
√21
2.
Page 3 of 10
MATH 105 921 Solutions to Probability Exercises
Problem 9. Are the following functions PDF’s?
(a) f(x) =
{12x2(x− 1) if 0 < x < 1
0 if otherwise.
Solution: f(x) is not a PDF because it fails the property that f(x) ≥ 0 for allx ∈ R. In fact, for any 0 < x < 1, we have that 12x2 > 0 (being an even power), butx− 1 < 0. So, for any 0 < x < 1,
f(x) = 12x2(x− 1) < 0.
For example, f
(1
2
)= −3
2< 0.
(b) f(x) =
{1− |x| if |x| ≤ 1
0 if otherwise..
Solution: f(x) is a PDF because it satisfies the following two properties:
• If |x| ≤ 1, then f(x) = 1 − |x| ≥ 0, and if |x| > 1, then f(x) = 0 ≥ 0, trivially.Thus, f(x) ≥ 0 for all x ∈ R.
• Observe that we may rewrite f(x) in more explicit form as:
f(x) =
1 + x if −1 < x < 0
1− x if 0 < x < 1
0 if otherwise.
So, ∫ ∞−∞
f(x) dx =
∫ 0
−1(1 + x) dx+
∫ 1
0
(1− x) dx
=
(x+
x2
2
)|0−1 +
(x− x2
2
)|10
= 0 + 1− 1
2+ 1− 1
2− 0 = 1.
Thus,
∫ ∞−∞
f(x) dx = 1.
(c) f(x) =
{π2
cos(πx) if |x| < 12
0 if otherwise.
Page 4 of 10
MATH 105 921 Solutions to Probability Exercises
Solution: f(x) is a PDF because it satisfies the following two properties:
• If |x| ≤ 1
2, then −π
2≤ πx ≤ π
2, which means that cos(πx) ≥ 0. Therefore,
f(x) =π
2cos(πx) ≥ 0 for |x| ≤ 1
2. If |x| > 1
2, then f(x) = 0 ≥ 0 trivially. Thus,
f(x) ≥ 0 for all x ∈ R.
• We have that:∫ ∞−∞
f(x) dx =
∫ 12
− 12
π
2cos(πx) dx
=1
2sin(πx) |
12
− 12
=1
2sin(π
2
)− 1
2sin(−π
2
)=
1
2+
1
2= 1.
Thus,
∫ ∞−∞
f(x) dx = 1.
(d) f(x) =
{12
if |x| ≤ 2
0 if otherwise.
Solution: f(x) is not a PDF because it fails to satisfy the property that
∫ ∞−∞
f(x) dx =
1. In fact, we have that: ∫ ∞−∞
f(x) dx =
∫ 2
−2
1
2dx
=1
2x |2−2
=1
2(2− (−2)) = 2.
Thus,
∫ ∞−∞
f(x) dx = 2 6= 1.
Problem 14. What is the CDF of the density function1
π(1 + x2)?
Page 5 of 10
MATH 105 921 Solutions to Probability Exercises
Solution: Recall that given a PDF f(x), we can find the CDF as follows:
F (x) =
∫ x
−∞f(t) dt
= lima→−∞
∫ x
a
1
π(1 + t2)dt
= lima→−∞
1
πarctan t |xa
= lima→−∞
1
πarctanx− 1
πarctan a
=1
πarctanx+
1
2.
Thus, the CDF of the given density function is F (x) =1
πarctanx+
1
2for any x ∈ R.
Problem 15. Show that p(x) =e−x
(1 + e−x)2on x ∈ R is a PDF.
Solution: To show that p(x) is a PDF, we need to verify the following two properties:
• We observe that e−x > 0 (being an exponential function), and (1+e−x)2 > 0 (beingan even power) for all x ∈ R. Thus,
p(x) =e−x
(1 + e−x)2> 0 for all x ∈ R.
• We have that: ∫ ∞−∞
p(x) dx =
∫ 0
−∞p(x) dx+
∫ ∞0
p(x) dx,
if both improper integrals on the right hand side converge. Using direct substitutionwith u = 1 + e−x, and du = −e−x dx, we get:∫
e−x
(1 + e−x)2dx =
∫−u−2 du = u−1 + C = (1 + e−x)−1 + C
Thus,∫ 0
−∞p(x) dx = lim
a→−∞
∫ 0
a
p(x) dx = lima→−∞
(1 + e−x)−1 |0a= lima→−∞
1
2− 1
1 + e−a=
1
2,∫ ∞
0
p(x) dx = limb→∞
∫ b
0
p(x) dx = limb→∞
(1 + e−x)−1 |b0= limb→∞
1
1 + e−b− 1
2=
1
2,∫ ∞
−∞p(x) dx =
1
2+
1
2= 1.
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MATH 105 921 Solutions to Probability Exercises
So,
∫ ∞−∞
p(x) dx = 1.
Therefore, p(x) is a PDF.
Problem 19. Let Z be a standard normal random variable (the classic ”bell curve”,given by the density:
φ(x) =1√2πe−
x2
2 .
Verify that E(Z) = 0.
Solution: We have that:
E(Z) =
∫ ∞−∞
xφ(x) dx =
∫ 0
−∞xφ(x) dx+
∫ ∞0
xφ(x) dx,
if both improper integrals on the right hand side converge. Using direct substitution
with u = −x2
2and du = −x dx to find an antiderivative of xφ(x), we get:∫
xφ(x) dx =
∫1√2πxe−
x2
2 dx =
∫− 1√
2πeu du = − 1√
2πeu + C = − 1√
2πe−
x2
2 + C
Thus, ∫ 0
−∞xφ(x) dx = lim
a→−∞
∫ 0
a
xφ(x) dx = lima→−∞
− 1√2πe−
x2
2 |0a= −1√2π,∫ ∞
0
xφ(x) dx = limb→∞
∫ b
0
xφ(x) dx = limb→∞− 1√
2πe−
x2
2 |b0=1√2π,
E(Z) = − 1√2π
+1√2π
= 0.
Therefore, E(Z) = 0, as desired.
Problem 20. Expectations (and variances) need not be finite. Let X be a Cauchy
random variable given by the PDF: p(x) =1
π(1 + x2), for x ∈ R.
(a) Prove that E(X) does not exist (i.e. the expectation is infinite).
Page 7 of 10
MATH 105 921 Solutions to Probability Exercises
Solution: We have that:
E(X) =
∫ ∞−∞
xf(x) dx =
∫ 0
−∞xf(x) dx+
∫ ∞0
xf(x) dx,
if both improper integrals on the right hand side converge. Using direct substitutionwith u = 1 + x2 and du = 2x dx to find an antiderivative of xf(x), we get:∫
xf(x) dx =
∫x
π(1 + x2)dx =
∫1
2πu−1 du =
1
2πln |u|+ C =
1
2πln(1 + x2) + C
Thus,∫ 0
−∞xf(x) dx = lim
a→−∞
∫ 0
a
xf(x) dx = lima→−∞
1
2πln(1 + x2) |0a= lim
a→−∞− 1
2πln(1 + a2) = −∞
Since
∫ 0
−∞xf(x) dx diverges, the improper integral
∫ ∞−∞
xf(x) dx also diverges. There-
fore, the expectations E(X) does not exist.
Problem 21. Let p(x) =1
βe−
xβ for 0 ≤ x < ∞, β > 0, an exponential random variable
with parameter β.
(a) Show that this density integrates to 1.
Solution: We have that:∫ ∞−∞
p(x) dx =
∫ ∞0
1
βe−
xβ dx
= limb→∞
∫ b
0
1
βe−
xβ dx
= limb→∞−e−
xβ |b0
= limb→∞−e−
bβ + 1 = 1.
Thus,
∫ ∞−∞
p(x) dx = 1, as desired.
(b) Calculate Pr(X ≥ 1), E(X) and StdDev(X), for the exponential random variable X.
Page 8 of 10
MATH 105 921 Solutions to Probability Exercises
Solution: We have that:
Pr(X ≥ 1) =
∫ ∞1
p(x) dx
=
∫ ∞1
1
βe−
xβ dx
= limb→∞
∫ b
1
1
βe−
xβ dx
= limb→∞−e−
xβ |b1
= limb→∞−e−
bβ + e−
1β
⇒ Pr(X ≥ 1) = e−1β .
Using integration by parts with u = x, du = dx, and dv = e−xβ dx, v = −βe−
xβ , we
find that the expected value is:
E(X) =
∫ ∞−∞
xp(x) dx
=
∫ ∞0
1
βxe−
xβ dx
= limb→∞
∫ b
0
1
βxe−
xβ dx
= limb→∞
(−xe−
xβ − βe−
xβ
)|b0
= limb→∞−be−
bβ − βe−
bβ + 0 + β (using L’Hopital’s Rule)
⇒ E(X) = β.
To find StdDev(X), we first compute Var(X) as follows:
Var(X) =
∫ ∞−∞
(x− E(X))2p(x) dx =
∫ ∞0
1
β(x− β)2e−
xβ dx
=
∫ ∞0
(1
βx2e−
xβ − 2xe−
xβ + βe−
xβ
)dx
= limb→∞
∫ b
0
(1
βx2e−
xβ − 2xe−
xβ + βe−
xβ
)dx
Using integration by parts twice on the first summand and integration by parts once
Page 9 of 10
MATH 105 921 Solutions to Probability Exercises
on the second summand, we get the following antiderivative of the integrand:
Var(X) = limb→∞
∫ b
0
(1
βx2e−
xβ − 2xe−
xβ + βe−
xβ
)dx
= limb→∞
(−β2e−
xβ − x2e−
xβ
)|b0
= lim→∞
(−β2e−
bβ − b2e−
bβ + β2 + 0
)(using L’Hopital’s Rule)
⇒ Var(X) = β2.
Therefore, StdDev(X) =√
Var(X) =√β2 = β, since β > 0.
Problem 24. Let X be a Laplace random variable given by the PDF: p(x) =1
2e−|x| for
x ∈ R.
(a) Verify that p(x) is in fact a valid PDF.
Solution: p(x) is a valid PDF because it satisfies the following two properties:
• For any real number x, we have that e−|x| ≥ 0 (being an exponential function).
Thus, p(x) =1
2e−|x| ≥ 0 for all x ∈ R.
• Observe that we may rewrite p(x) in more explicit form as:
p(x) =
{12ex if x < 0
12e−x if 0 ≤ x
So, ∫ ∞−∞
p(x) dx =
∫ 0
−∞
1
2ex dx+
∫ ∞0
1
2e−x dx (if both integrals converge)
= lima→−∞
∫ 0
a
1
2ex dx+ lim
b→∞
∫ b
0
1
2e−x dx
= lima→−∞
1
2ex |0a + lim
b→∞−1
2e−x |b0
= lima→−∞
1
2(1− ea) + lim
b→∞−1
2
(e−b − 1
)=
1
2+
1
2= 1.
Thus,
∫ ∞−∞
p(x) dx = 1.
Page 10 of 10