Worksheet 3 – Solutions
Chapter (3): Random Variables and Probability Distributions
Chapter (4): Mathematical Expectation
Q−1: Let X be a random variable giving the number of heads minus the number of tails in three tosses of a coin.
Assume that the coin is biased so that a head is twice as likely to occur as a tail.
a) List the elements of the sample space S for the three tosses of the coin and to each sample point assign a
value x of X.
S
X
HHH
3
HHT
1
HTH
1
THH
1
HTT
−1
THT
−1
TTH
−1
TTT
−3
b) Find the probability distribution of the random variable X.
P(H) = 2/3 & P(T) = 1/3
P(X = −3) = P(TTT) = (1/3)(1/3)(1/3) = 1/27
P(X = −1) = P(HTT) + P(THT) + P(TTH) = 3(1/3)(1/3)(2/3) = 6/27= 2/9
P(X = 1) = P(HHT) + P(HTH) + P(THH) = 3(1/3)(2/3)(2/3) = 12/27 = 4/9
P(X = 3) = P(HHH) = (2/3)(2/3)(2/3) = 8/27
x
−3
−1
1
3
P(X = x)
1/27
2/9
4/9
8/27
c) Find P(−1 ≤ X ≤ 3).
P(−1 ≤ X ≤ 3) = P(X = −1) + P(X = 1) + P(X = 3) = 26/27.
___________________________________________________________________________________
Q–2: Suppose that the number of cars X that pass through a car wash between 1:00 P.M. and 2:00 P.M. on any
sunny Friday has the following probability distribution:
4
5
x
1/12
1/12
P(X = x) = f (x)
a) Verify that f (x) is a probability mass function.
9
1
1
1
1
1
1
 f x   12  12  4  4  6  6
6
1/4
7
1/4
8
1/6
9
1/6
1
x4
b) Find P(X < 6) and P(5 ≤ X ≤ 7).
P X  6 
5
 f x   12  12  6
1
1
1
x4
P 5  X  7  
7
 f x   12  4  4  12
1
1
1
7
x 5
c) Find the expected number of cars that pass through a car wash during this particular time period.
EX  
9
 xf x   12  12  4  4  6  6
4
5
6
7
8
x4
9

82
 6.8 cars
12
___________________________________________________________________________________

k ,
Q–3: If f  x   
0,
a) 2
1
2 is a density function, then k =
otherwise
0 x
(b) 4
(c) 1
(d) 0.5
(e) None of the above
_________________________________________________________________________________
Q−5: The waiting time, in hours, between successive speeders spotted by a radar unit is a continuous random
variable with probability density function:
8e 8 x , x  0
f x   
elsewhere
0,
a) Find the cumulative distribution function of X.
F x  
x

f t dt 
0
x
 8e
0
 8t
dt   e  8t
x
0
 1  e 8 x .
x0
0,
F x   
.
8 x
1

e
,
x

0

b) Find the probability of waiting less than 12 minutes between successive speeders.
P X  0.2  F 0.2  1  e 1.6  0.7981.
c) Find the probability of waiting from 12 to 24 minutes between successive speeders.
P0.2  X  0.4  F 0.4  F 0.2  e 1.6  e 3.2  0.1611.
___________________________________________________________________________________
Q−6: The density function of the continuous random variable X, the total number of hours, in units of 100
hours, that a family runs a vacuum cleaner over a period of one year, is given as
0  x 1
 x,

f x   a  x, 1  x  2
.
0,
elsewhere

a) Find the constant a.

f  x dx  1 



1
2
0
1
 xdx   a  x dx  1 
1 2
x
2
1
0
2
1


  ax  x 2   1
2

1
1
1

 2a  2   a    1  a  2.
2
2

b) Find the average number of hours per year that families run their vacuum cleaners.

2
1
1
1 

E  X     xf  x dx   x dx   x2  x dx  x 3   x 2  x 3 
3 0 
3 1

0
1
1
2
2
1 
8  1
  4    1    1.
3 
3 
3
Therefore, the average number of hours per year is 100 hours.

c) Find the variance and standard deviation of X.


  x f x dx   x
1
E X2 
2

2
3
dx   x 2 2  x dx 
0
1
1 4
x
4
1
0
2
1
2

  x3  x4 
4
3
1
1  16
 2 1 7
 
 4      .
4  3
 3 4 6
7
1
Var  X   E X 2  E  X    1 
6
6
Therefore, the variance is 16.67 hours and the standard deviation is 4.08 houts.


___________________________________________________________________________________
3
x, 0  x  1

Q–7: Consider the density function: f  x    2

elsewhere
0,
a) Find P(0.3< X < 0.6).
3 0.6
0.6
3
P0.3  X  0.6  
x dx  x 2
2
0.3
 0.3004
0.3
b) Find the mean of the random variable X.
5 1
3
3
3

  E  X    x
x  dx   x 2 dx  x 2
2
20
5

0 
1
1
3

0
3
 0.6
5
c) Find the variance of the random variable X.
 
EX
2
7 1
3
3
3
3

 x 
x  dx   x 2 dx  x 2 
20
7 0 7
2

0
1
1
5
2
 2  E X 2    2  
3
7
9
12

 0.069
25 175
___________________________________________________________________________________
Q-8:
___________________________________________________________________________________
Q-9: