Other Important Random Variables - Random Signals and Processes Primer with MATLAB - page 344

The mean value of Y is:

1
3

k 5 k

EfYg ¼ X 5 g ¼ X

5 5

5
k

2
3

e 2 k PfX ¼ k ; e 2 k

k¼ 0 k¼ 0

e 2 1
3

k 5 k

¼ X

5

2
3

5
k

:

(5.257)

k¼ 0

Comparing ( 5.112 ) with ( 5.257 ), we can write:

5

e 2
3 þ

2
3

EfYg ¼ ¼ 0 : 1827 (5.258)

Exercise 5.14 If it is known that 20% of students did not pass an exam, find a
probability that out of five randomly chosen students:

(a) One student did not pass exam.
(b) All five students passed the exam.
(c) At least three students passed the exam.

Answer Denote as a binary event A that a student did not pass the exam.
The probability of event A is defined as:

PfAg ¼ 0 : 2 : (5.259)

The probability that k out of n students did not pass the exam can be calculated
using the Bernoulli formula.

(a) The probability that one out of five randomly chosen students did not pass the
exam is:

0

5

5
1

!
ð 5 1 Þ ! 1 !

8 4 8 4

Pf 1

5 g ¼

:

2 0

: ¼ :

2 0

: ¼ 0 : :

; 0 4096 (5.260)

(b) The probability that all five randomly chosen students passed the exam (that is
k ¼ 0 students failed) is:

0

5

5
0

!
ð 5 0 Þ !

2 0 8 5 8 5 8 5

Pf 0 5 g ¼ : 0 : ¼ 0 : ¼ 0 : ¼ 0 : 3277 : (5.261)

;

0

!

(c) In this case, 3, 4, or 5 students passed exam. This means that 2, 1, or 0 students
did not pass the exam, or k 2:

0 : 2 k

Pfk 2 ; 5 g¼ X

2

5
k

0 : 8 5 k

k¼ 0

5 5 5

!
ð 5 0 Þ ! 0 !

!
ð 5 1 Þ ! 1 !

!
ð 5 2 Þ ! 2 !

0 : 8 5 0 : 2 0 : 8 4 0 : 2 2 0 : 8 3 ¼ þ þ

(5.262) ¼ 0 : 3277 þ 0 : 4096 þ 0 : 2048 ¼ 0 : 9421 :