lOMoAR cPSD| 46836766 SEMINAR 11
1. Which of the following would you treat as continuous random variables?
a. The number of people who arrive for a particular charter 昀氀 ight.
b. The number of mm of rain that falls on London on a particular day.
c. The number of sta 昀昀 in a small o 昀케 ce who are o 昀昀 sick on a particular day.
d. The proportion of phone calls from a particular payphone on a particular day that take longer than two minutes.
e. The duration of a phone call from a payphone.
f. The number of telephone units on a telephone card that a particular call uses.
2. A random variable, Y, can take values between 0 and 0.5 and has the
PDF f ( y)=8 y.
a. Calculate the probability that Y is larger than 0.25.
b. Calculate the probability that Y is smaller than 0.1.
c. Calculate the probability that Y lies between 0.1 and 0.25.
3. A random variable X follows the continuous uniform distribution with a lower bound of 5 and an upper bound of 35.
a. What is the height of the density function f ( x )?
b. What are the mean and the standard deviation for the distribution?
c. Calculate Pr ( X>10).
4. A local authority is responsible for a stretch of road 3km long through a town. A gas main
runs along the length of the road. The gas company has requested permission to dig up
the road in one place but has neglected to tell the local authority exact location.
a. Let X be the distance of the roadworks from one end of the road. Sketch the PDF of X.
b. What is f ( x )?
c. The stretch of road between 1.5km and 2.75km from one end goes through the town
centre and roadworks there would cause severe disruption. What is the probability that this happens? lOMoAR cPSD| 46836766
5. Suppose the average price of electricity for a New England customer follows the
continuous uniform distribution with a lower bound of 12 cents per kilowatt-hour and
an upper bound of 20 cents per kilowatthour.
a. Calculate the average price of electricity for a New England customer.
b. What is the probability that a New England customer pays less than 15.5 cents per kilowatt-hour?
c. A local carnival is not able to operate its rides if the average price of electricity is more
than 14 cents per kilowatt-hour. What is the probability that the carnival will need to close?
6. Given that Z is a standard normal random variable, compute the following probabilities.
a. Pr(0≤Z≤1.96)
b. Pr(1.52≤ Z≤1.96)
c. Pr(−1.52≤Z≤1.96)
d. Pr ( Z>4 )
7. For the standard normal variable Z, 昀椀 nd the z values that satisfy the following
probability statements. a. Pr ( Z≤ z)=0.6808
b. Pr ( Z>z)=0.0212
c. Pr ( Z≤ z)=0.90
d. Pr (−z ≤Z ≤z)=0.95
e. Pr ( Z≤ z)=0.0643
8. Assume X follows a normal distribution with mean 100 and standard deviation 15, i.e. X
N (100,152).What is Pr ( X<85) and Pr ( X>130)?
9. Scores on a management aptitude exam are normally distributed with a mean of 72 and a standard deviation of 8.
a. What is the probability that a randomly selected manager will score above 60?
b. What is the probability that a randomly selected manager will score between 68 and 84?
c. What is the lowest score that will place a manager in the top 10% (90 th percentile) of the distribution? lOMoAR cPSD| 46836766
d. What is the highest score that will place a manager in the bottom 25 %
(25 th percentile) of the distribution?
10. A professional basketball team averages 80 points per game with a standard deviation
of 10 points. Assume points per game follow the normal distribution.
a. What is the probability that a game’s score is between 60 and 100 points?
b. What is the probability that a game’s score is more than 100 points? If there are 82
games in a regular season, in how many games will the team score more than 100 points?