1. Reference: (Material source and pages)
2018/8/7
Unit 10 Expectation
IT Disicipline
ITD1111 Discrete Mathematics & Statistics STDTLP

2. 10.1 Mathematical expectation
In each trial of an experiment, a random variable X may assume any value in the range space R. In the long-run, an average and a variance may be associated with X.
IT Disicipline
ITD1111 Discrete Mathematics & Statistics STDTLP

3. ITD1111 Discrete Mathematics & Statistics STDTLP
Example (1/4)
The length in meters, X, of metal bars produced in a machine is such that P(X < 4.5) = 0.1 P(4.5  X  5) = 0.7 P(X > 5) = 0.2.
IT Disicipline
ITD1111 Discrete Mathematics & Statistics STDTLP

4. ITD1111 Discrete Mathematics & Statistics STDTLP
Example (2/4)
If 4.5  X  5, a bar will be accepted by the customers with a profit of $52.
If X < 4.5, a bar will be discarded with a loss of $20.
If X > 5, a bar will be cut to the specified length and the profit will be reduced to $48.
Find the expected profit.
IT Disicipline
ITD1111 Discrete Mathematics & Statistics STDTLP

5. ITD1111 Discrete Mathematics & Statistics STDTLP
Example (3/4)
Consider N bars be produced and
let A be the no. of bars accepted,
R be the no. of bars rejected, and
C be the no. of bars that have to be cut.
Average profit P = =
IT Disicipline
ITD1111 Discrete Mathematics & Statistics STDTLP

6. ITD1111 Discrete Mathematics & Statistics STDTLP
Example (4/4)
 = = 44 The expected profit is therefore $44.
IT Disicipline
ITD1111 Discrete Mathematics & Statistics STDTLP

7. 10.2 Expected value (definition)
In general, let X be a discrete r.v. with possible values x1, x2, …, xn and corresponding probabilities p1, p2, …, pn. The expectation of X is defined as E(X) = = x1p1 + x2p2 + … + xnpn . The expectation of X is also called the expected value of X.
IT Disicipline
ITD1111 Discrete Mathematics & Statistics STDTLP

8. ITD1111 Discrete Mathematics & Statistics STDTLP
Example
The random variable X has the distribution shown below x
P(X = x)
The expected value of X, E(X)
= (1)(0.3) + (2)(0.2) + (3)(0.4) + (4)(0.1)
= 2.3
IT Disicipline
ITD1111 Discrete Mathematics & Statistics STDTLP

9. ITD1111 Discrete Mathematics & Statistics STDTLP
Example (1/2)
The random variable X has a probability distribution given by P(X = x) = , x = 1, 2, 3, 4, 5. (a) Find c. (b) Find E(X)
IT Disicipline
ITD1111 Discrete Mathematics & Statistics STDTLP

10. ITD1111 Discrete Mathematics & Statistics STDTLP
Example (2/2)
(a) c = (b) E(X) = = 5c =
IT Disicipline
ITD1111 Discrete Mathematics & Statistics STDTLP

11. ITD1111 Discrete Mathematics & Statistics STDTLP
Example
In a game, a player tosses 3 fair coins. He wins $10 if 3 heads occur, $5 if 2 heads occur, $2 if only 1 head occurs and losses $15 if no heads occur. What is his expected gain? His expected gain = 10(1/8)+5(3/8)+3(3/8)-15(1/8) = 2 (dollars)
IT Disicipline
ITD1111 Discrete Mathematics & Statistics STDTLP

12. 10.3 Function of a random variable
If X is a discrete random variable with probability distribution p(x) and if g(x) is a real-valued function of X, then the expected value of g(X) is defined as
E[g(X)] =
IT Disicipline
ITD1111 Discrete Mathematics & Statistics STDTLP

13. ITD1111 Discrete Mathematics & Statistics STDTLP
Example (1/3)
A salesman is employed by a computer manufacturer to sell PC’s. The salary he gets in a day is calculated by the formula
g(x) = x
where x is the number of PC’s he sells in that day.
IT Disicipline
ITD1111 Discrete Mathematics & Statistics STDTLP

14. ITD1111 Discrete Mathematics & Statistics STDTLP
Example (2/3)
Assume that in each day he may sell zero to four PC’s with probabilities listed in the table below:
Number of PC’s, x
Probability, p(x)
IT Disicipline
ITD1111 Discrete Mathematics & Statistics STDTLP

15. ITD1111 Discrete Mathematics & Statistics STDTLP
Example (3/3)
Let X be the number of PC’s sold in a day.
His expected daily salary is
E[g(X)] = g(0)p(0) + g(1)p(1) + g(2)p(2) +
g(3)p(3) +g(4)p(4)
= (90)(0.05) + (90+60)(0.2) +
(90+120)(0.4) + (90+180)(0.2) +
(90+240)(0.15)
= 222 (dollars)
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ITD1111 Discrete Mathematics & Statistics STDTLP

16. ITD1111 Discrete Mathematics & Statistics STDTLP
Expectation of aX + b
It can be proved that, for any real numbers a and b,
E(aX + b) = aE(X) + b .
IT Disicipline
ITD1111 Discrete Mathematics & Statistics STDTLP

17. ITD1111 Discrete Mathematics & Statistics STDTLP
Example
In Example 7.3-5,
expected daily salary
= E(90+60X)
= E(X)
= [(0)(0.05) + (1)(0.2) + (2)(0.4)
+ (3)(0.2) + (4)(0.15)]
= 222 (dollars)
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ITD1111 Discrete Mathematics & Statistics STDTLP

18. 10. 4 Variance of a random variable
The variance of a random variable X is defined as
V(X) = 2 = E[(X –)2] ,
where  = E(X).
The standard deviation, , of a random variable X is defined
 =
IT Disicipline
ITD1111 Discrete Mathematics & Statistics STDTLP

19. ITD1111 Discrete Mathematics & Statistics STDTLP
Example (1/2)
The probability distribution of the random variable X is shown below: X
P(X = x)
Find E(X) and V(X).
IT Disicipline
ITD1111 Discrete Mathematics & Statistics STDTLP

20. ITD1111 Discrete Mathematics & Statistics STDTLP
Example (2/2)
E(X) =
= 1(0.1) + 2(0.5) + 3(0.4)
= 2.3
V(X) = E[(X –)2]
= (1 – 2.3)2(0.1) + (2 – 2.3)2(0.5) + (3 – 2.3)2(0.4)
= 0.41
IT Disicipline
ITD1111 Discrete Mathematics & Statistics STDTLP

21. ITD1111 Discrete Mathematics & Statistics STDTLP
Example (1/2)
Daily sales records for a shop selling electric appliances show that it will sell zero, one, two or three air-conditioners with the probabilities:
Number of Sales
Probability
Calculate the expected value, variance and standard variation for daily sales.
IT Disicipline
ITD1111 Discrete Mathematics & Statistics STDTLP

22. ITD1111 Discrete Mathematics & Statistics STDTLP
Example (2/2)
Expected value
= (0)(0.5) + (1)(0.3) + (2)(0.15) + (3)(0.05)
= 0.75
Variance
= ( )2(0.5) + (1 – 0.75)2(0.3) +
(2 – 0.75)2(0.15) + (3 – 0.75)2(0.05) =
Standard deviation = =
IT Disicipline
ITD1111 Discrete Mathematics & Statistics STDTLP