Blekinge Institute of Technology
School of Engineering
Department of Mathematics and Science
CJO
MSA002
Introductory Exercises in Probability
Chapter 2
1. Are the functions below proper cdf:s? Motivate!
a) F(x) = e-x, x>0.
b) F(x) = ex, x>0.
c) F(x) = 2x, 0<x<1/2.
2. For the exponential distribution, the density is f X ( x)  e x , x  0 (0 elsewhere ) .
a) Find the cdf FX(x).
b) Find the probability that X is between 2 and 3.
3. The uniform distribution has density
1
f X ( x) 
, a  x  b.
ba
a) Calculate P( X  (a  b )/2) by integrating the density.
b) Find the cdf and solve a) by using the cdf.
c) Find the mean and the variance of X.
4. A random variable has the density f X ( x)  ce  x , 0  x  10. Find the value of the
constant c.
5. Find the mean and variance of X+Y if E(X) = E(Y) = m, V(X) = a2, V(Y) = b2. Assume
that X and Y are independent.
6. Find the mean, variance and standard deviation of cX+dY with the X and Y from the
preceding exercise.
7. The random variables X i , i  1,..., n follow the normal (Gaussian) distribution, each with
mean 10 and standard deviation a. Calculate the mean, variance and standard deviation of
the arithmetic mean X   X 1  ...  X n  / n.
8. The random variable X is standardised Gaussian, i.e. the mean equals zero and the
standard deviation equals 1. Calculate the following probabilities.
a) P( X  1.2) b) P( X  1.2)
c) P( X  0.5) d) P( X  1)
9. The random variable X is the same as in the preceding exercise. Find a value of the
constant c such that
a) P( X  c)  0.95
b) P( X  c)  0.99
10. A random variable X is Gaussian (normal) with mean 5 and standard deviation 2.
a) Calculate P( X  7).
b) Calculate P( X  2).
11. The length X of female students at BTH is assumed to follow a normal distribution with
mean 168 cm and standard deviation 7 cm. Calculate
a) P(168<X<182)
b) the probability that all students in a group of five are shorter than 175 cm
c) the probability that the mean length in this group exceeds 175 cm
12. An engineer is testing ten new electronic devices. It is known that the probability that a
new device works is 0.95. Find the probability that
a) all work
b) exactly nine work
c) nine or less work
13. A telecommunication engineer investigates the traffic in a certain network. The number of
phone calls is assumed to be Poisson with mean m = 5 calls/minute. Find the probability
that
a) there will no phone call during a certain minute
b) there will be at least one phone call during a certain minute
Chapter 3
14. Find the mean E(X) of a variable X with density f X ( x)  e x , x  0 (exponential
distribution).
15. A student has five coins in his wallet: one one-crown, three five-crowns and one tencrown coin. He chooses a coin randomly. Calculate the mean, variance and standard
deviation of its value X.
16. A random variable X is uniformly distributed between 0 and 2, i.e.
f X ( x)  1/ 2, 0  x  2. Calculate the cdf of Y  X .
17. The random variable X is exponentially distributed, i.e. f X ( x)  e x , x  0.
Calculate the cdf and the density of Y  X 2 .
18. For the variable X in the preceding exercise, calculate the cdf and the density of Z  ln X .
19. The random variables X i , i  1,..., n are independent and exponentially distributed.
Calculate the cdf of Y  min( X 1 ,..., X n ) and Z  max( X 1 ,..., X n ).
20. Two components are working in parallel. Each component has an exponential density
with parameter  . Calculate the cdf of the lifetime of the system (Y).
Answers
1. a) No, the function is decreasing. b) No, the function attains values larger than 1.
c) Yes.
2. a) 1  e x , x  0. b) e 2  e 3
xa
, a  x  b. Insertion gives 0.5.
3. a) 0.5 b)
ba
c) E(X) = (a+b)/2. V(X) = E(X2)- (E(X))2 = (b-a)2/12.
4. c = 1/(1-e-10)
5. E(X+Y) = 2m, V(X+Y) = a2+b2
6. E(X+Y) = (c+d)m, V(X+Y) = c2a2+d2b2,  X Y  c 2 a 2  d 2 b 2 .
7. E(X) = 10, V(X) = a2/n,   a / n.
8. a)  (1.2)  0.8849. b) 1   (1.2)  0.1151. c)  (0.5)  1  (0.5)  0.3085.
d) 1   (1)   (1)  0.8413.
9. a) 1.645 b) 2.326
7 5
10. a) 
   (1)  0.8413
 2 
 25
b) 
   1.5  1   (1.5)  1  0.9332  0.0668.
 2 
11. a)
P(168  X  180)  P((168  168) / 7  ( X  168) / 7  (182  168) / 7)  (2)  (0) 
 0.9772  0.5  0.4772
b) P( X  1755  (1)5  0.84135  0.4215.
c)
X  N (168,7 / 5 )  P( X  175)  P( X  168 5 / 7  175  168 5 / 7) 
P(Y  5)  1  P(Y  5 )  1   ( 5 )  1  0.9874  0.0126.
10 
10
9
12. a) 0.95  0.599 b)  0.95 0.05  0.315 c) 1-answer in a) = 0.401.
9
5
13. a) e
50
 0.0067 b) 0.9933
0!
14. E ( X )  1 /  .
15. E(X) =  xp(x)  1  0.2  5  0.6  10  0.2  5.2. E(X2) = 35.2.
V(X) = E ( X 2 )  E ( X )2  8.16.   V ( X )  2.86.
16. FY ( x)  x 2 / 2, 0  x  2.
17. FY ( x)  1  e 
, x  0,
x
18. FY ( x)  1  e e , x  0,
x
f Y ( x) 

2 x
e
 x
f Y ( x)  e x e e , x  0.
x


19. FY ( x)  1  e nx , FZ ( x)  1  e x , x  0.


20. FY ( x)  1  e x , x  0.
2
, x  0.
n