CEGEP CHAMPLAIN - ST. LAWRENCE
201-203-RE: Integral Calculus
Patrice Camiré
Problem Sheet #17
Probability and Calculus
1. Show that each function is a probability density function on the given interval.
(a) f (x) = e−x , [0, ∞).
(c) f (x) = 6x(1 − x) , [0, 1].
(b) f (x) = xe−x , [0, ∞).
(d) f (x) =
ln(x)
, [1, ∞).
x2
2. Find the value of k > 0 such that f (x) is a probability density function on the given interval.
√
(a) f (x) = kx 1 − x , [0, 1].
(b) f (x) = √
kx
, [0, 2].
x2 + 1
(c) f (x) =
x2
k
, [0, ∞).
+ 4x + 3
(d) f (x) = k sin3 (x) , [0, π].
3. For each probability density function, find the mean, the variance and the standard deviation of
the corresponding random variable.
(a) f (x) = 6x(1 − x) , [0, 1].
(d) f (x) = 4
(b) f (x) = 60x3 (1 − x)2 , [0, 1].
(c) f (x) =
3
, [1, ∞).
x4
(e) f (x) =
ln(x)
, [1, ∞).
x3
sin(x)
, [0, π].
2
4. Data has shown that the time T a person waits at the bank before being serviced by a teller is
exponentially distributed with probability density function f (t) = 0.1e−0.1t , where t is in minutes.
(a) What is the probability that a person waits less than five minutes before seeing a teller?
(b) What is the probability that a person waits more than fifteen minutes before seeing a teller?
(c) What is the probability that a person waits between eight and twelve minutes before seeing a
teller?
(d) What is the expected waiting time?
5. Data has shown that the waiting time T at a bus stop on Sunday is exponentially distributed with
probability density function f (t) = 0.05e−0.05t , where t is in minutes.
(a) What is the probability that a person waits more than 25 minutes?
(b) What is the probability that a person waits less than 5 minutes?
(c) Suppose that you arrive at the bus stop at noon and agreed to meet a friend at the mall at
12:30. If the bus takes on average 20 minutes to get to the mall, do you expect to be late?
6. (Optional) You are at a fair and decide to play a computerized game of chance. It costs $5 to
2
2
play and your gain X is distributed according to the probability density function f (x) = √ e−x ,
π
[0, ∞). The appeal of this game is that there is no upper bound on the player’s gain, but is it really
such a great game? Let us investigate!
(a) What is the expected gain of this game?
(b) Use Maple to find the probability of making money if you play this game?
(c) Looking at yours answers from (a) and (b), should you play this game?
Answers
Z
1. Make sure that in each case, f (x) ≥ 0 on [a, b] and
b
f (x) dx = 1.
a
2. (a) k =
15
4
(b) k = √
1
5−1
√
1 2
1
5
3. (a) µ = , σ =
and σ =
.
2
20
10
√
4 2
3
6
(b) µ = , σ =
and σ =
.
7
98
14
4. (a) P (T ≤ 5) ≈ 0.393
(b) P (T ≥ 15) ≈ 0.223
2
ln(3)
3
(d) k =
4
(c) k =
√
3 2 3
3
(c) µ = , σ = and σ =
.
2
4
2
(d) µ = 4, σ 2 = ∞ and σ = ∞.
π
π2
(e) µ = , σ 2 =
− 2 and σ =
2
4
(c) P (8 ≤ T ≤ 12) ≈ 0.148
(d) µ = E(T ) = 10 minutes.
5. (a) P (T ≥ 25) ≈ 0.287
(b) P (T ≤ 5) ≈ 0.221
(c) Yes, since you expect to arrive at the mall at 12:40.
6. (a) µ = E(X) ≈ $0.57. You are therefore expected to lose ≈ $4.43.
(b) Using Maple, we find that P (X ≥ 5) ≈ 1.54 × 10−12 .
(c) No! This game is a scam.
√
π2 − 8
.
2