MACT 3224/317: Assignment N ◦3
Rahmatullah Imon, Khouzeima Moutanabbir, Abd-Elnasser Saad and Noha Youssef∗
Fall 2015
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• Due Wednesday, November 11, 2015 by 3:00 pm.
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∗ The
Department of Mathematics and Actuarial Science, The American University in Cairo.
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• Question 1:
Engineers observe that about 90% of graphite samples fracture within 5 hours when subjected to a
certain stress.
(a) If the time to fracture is modeled with an exponential distribution, what would be a suitable value
for the parameter λ?
(b) Use the Exponential model to find the probability that a fracture occurs within 3 hours
• Question 2:
A new car that is gas- and electric- powered hybrid has recently hit the market. The distance traveled
on 1 gallon of fuel is normally distributed with a mean of 65 miles and a standard deviation of 4 miles.
Find the probability that
(a) the car travels less than 60 miles per gallon;
(b) the car travels between 55 and 70 miles per gallon.
• Question 3:
The following density function describes the random variable Y
y
25 , if 0 < y < 5,
10−y
f (y) =
, if 5 ≤ y ≤ 10,
25
0, elsewhere .
(a) Find the distribution function of Y .
(b) Compute the probabilities P (4 ≤ Y ≤ 8), P (3 ≤ Y ) and P (Y < 9 | Y ≥ 2).
(c) Find the expected value and the variance of Y .
• Question 4
Suppose the lifetime in days, X, of a car battery has the following probability density function
f (x) =
Cx2 (100 − x)2 , if 0 ≤ x ≤ 100,
0, elsewhere .
(a) Find the value of C.
(b) Find the probability that the battery lives more than 60 days.
• Question 5:
Let Y be a random variable with the following density distribution function
fY (y) =
4 −y 2 −2y
e + e
,
5
5
for y ≥ 0
(a) Find the probability that Y is less than 1 given that Y is greater than 0.5.
(b) Calculate the expected value of 2015 − 6Y 2 + 24Y .
• Question 6:
The amount of mineral water consumed by a person per day on the job is normally distributed with
mean 19 ounces and standard deviation 5 ounces. A company supplies each employee with x ounces
of mineral water daily.
(a) Assuming the x = 25, find the probability that the mineral water supplied by the company will
not satisfy the water demanded by the employee.
(b) Find the minimal value of x such that the mineral water supplied by the company satisfies the
water demanded by the employee with probability at least 10%.
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• Question 7:
Suppose the proportion X of surface area in a randomly selected quadrat that is covered by a certain
plant has a beta distribution with α = 3 and β = 2
(a) Find the probability that this proportion is no less than 50%.
(b) Find the probability that this proportion exceeds 20% given that it is less than 50%.
• Question 8:
Let Y be a random variable that follows a Gamma distribution with parameters α = 3 and β = 2.
Find the probability that Y is less than its expected value.
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