Course: Mathematics-I (CDA101), Topic- Random Variables
Tutorial Sheet #3
1. Classify the following random variables as discrete or continuous:
X: the number of automobile accidents per year in Virginia.
Y : the length of time to play 18 holes of golf.
M: the amount of milk produced yearly by a particular cow.
N: the number of eggs laid each month by a hen.
P: the number of building permits issued each month in a certain city.
Q: the weight of grain produced per acre.
2. Find and plot the CDFs corresponding to each of the following PDFs:
(i) 1
b−a
:a≤x≤b
f (x) =
0 : otherwise
(ii)
x :0≤x≤1
f (x) = 2−x :1≤x≤2
0 : otherwise
3. If the distribution function of X is given by
0 :b<0
1/2 :0≤b<1
3/5 :1≤b<2
F (b) =
4/5 :2≤b<3
9/10 : 3 ≤ b < 3.5
1 : b ≥ 3.5,
calculate the probability mass function of X.
4. Prove that E[X 2 ] ≥ (E[X])2 . When do we have equality?
5. The Department of Energy (DOE) puts projects out on bid and generally estimates
what a reasonable bid should be. Call the estimate b. The DOE has determined
that the density function of the winning (low) bid is
5
8b
: 2b
5
< y < 2b
f (y) =
0 : otherwise
Find F (y) and use it to determine the probability that the winning bid is less than
the DOEs preliminary estimate b.
6. The waiting time, in hours, between successive speeders spotted by a radar unit is
a continuous random variable with cumulative distribution function
0: x<0
F (x) = −8x
1−e x ≥ 0.
Find the probability of waiting less than 12 minutes between successive speeders
(a) using the cumulative distribution function of X;
(b) using the probability density function of X.
1
Tutorial Sheet #3
1. Classify the following random variables as discrete or continuous:
X: the number of automobile accidents per year in Virginia.
Y : the length of time to play 18 holes of golf.
M: the amount of milk produced yearly by a particular cow.
N: the number of eggs laid each month by a hen.
P: the number of building permits issued each month in a certain city.
Q: the weight of grain produced per acre.
2. Find and plot the CDFs corresponding to each of the following PDFs:
(i) 1
b−a
:a≤x≤b
f (x) =
0 : otherwise
(ii)
x :0≤x≤1
f (x) = 2−x :1≤x≤2
0 : otherwise
3. If the distribution function of X is given by
0 :b<0
1/2 :0≤b<1
3/5 :1≤b<2
F (b) =
4/5 :2≤b<3
9/10 : 3 ≤ b < 3.5
1 : b ≥ 3.5,
calculate the probability mass function of X.
4. Prove that E[X 2 ] ≥ (E[X])2 . When do we have equality?
5. The Department of Energy (DOE) puts projects out on bid and generally estimates
what a reasonable bid should be. Call the estimate b. The DOE has determined
that the density function of the winning (low) bid is
5
8b
: 2b
5
< y < 2b
f (y) =
0 : otherwise
Find F (y) and use it to determine the probability that the winning bid is less than
the DOEs preliminary estimate b.
6. The waiting time, in hours, between successive speeders spotted by a radar unit is
a continuous random variable with cumulative distribution function
0: x<0
F (x) = −8x
1−e x ≥ 0.
Find the probability of waiting less than 12 minutes between successive speeders
(a) using the cumulative distribution function of X;
(b) using the probability density function of X.
1
