Chapter 3
Chapter 3
Chapter 3
Mathematical Expectation
if X is discrete, and
Z +∞
µ = E(X) = xf (x)dx
−∞
if X is continuous.
Example 3.2: Let X be the random variable that denote the life in hours of a
certain electronic device. The probability distribution function is given by
(
20,000
x3
, x > 100
f (x) =
0, elsewhere
1
Class Notes on
3.1. MATHEMATICAL EXPECTATION Applied Probability and Statistics ECEG-342
if X is discrete, and
Z +∞
µg(X) = E[g(X)] = g(x)f (x)dx
−∞
if X is continuous
Example 3.3: Suppose that the number of cars, X, that park in a particular area
between 10:00 A.M. and 12:00 A.M. on any Friday has the following probability
distribution:
x 4 5 6 7 8 9
1 1 1 1 1 1
P (X = x) 12 12 4 4 6 6
Let g(X) = 41 X − 0.1 represent the amount of money in Birr paid to the attendant
by the park manager. Find the attendant’s expected earnings for this particular
time period.
E(aX + b) = aE(X) + b
Murad Ridwan, 2 of 6
Dep. of Electrical & Computer Engineering
AiOT, Addis Ababa University.
Jul 2010.
Class Notes on
3.2. VARIANCE AND STANDARD DEVIATION Applied Probability and Statistics ECEG-342
E(XY ) = E(X)E(Y )
if X is discrete, and
Z +∞
2 2
σ = E[(X − µ) ] = (x − µ)2 f (x)dx
−∞
Murad Ridwan, 3 of 6
Dep. of Electrical & Computer Engineering
AiOT, Addis Ababa University.
Jul 2010.
Class Notes on
3.3. COVARIANCE Applied Probability and Statistics ECEG-342
2 2 2
If a = 1, σX+b = σX and if b = 0, σaX = a2 σ X
2
.
Theorem 9. If X and Y are independent random variables, then
Var(aX ± bY ) = a2 Var(X) + b2 Var(Y ) or 2
σaX±bY = a2 σX
2
+ b2 σY2
Theorem 10. If a random variable X has mean µ and variance σ 2 , then the
corresponding random variable
X −µ
Z=
σ
has mean 0 and variance 1.
if X is continuous.
3.3 Covariance
The concept of variance given for one variable can be extended to two or
more variables. Thus, for example, if X and Y are two continuous random
variables having joint density f (x, y) then, the mean and variances are
Z +∞ Z +∞
µX = E(X) = xf (x, y)dxdy
−∞ −∞
Z +∞ Z +∞
µY = E(Y ) = yf (x, y)dxdy
−∞ −∞
Z +∞ Z +∞
2 2
σX = E[(X − µX ) ] = (x − µX )2 f (x, y)dxdy
−∞ −∞
Z +∞ Z +∞
σY2 = E[(Y − µY )2 ] = (y − µY )2 f (x, y)dxdy
−∞ −∞
Murad Ridwan, 4 of 6
Dep. of Electrical & Computer Engineering
AiOT, Addis Ababa University.
Jul 2010.
Class Notes on
3.3. COVARIANCE Applied Probability and Statistics ECEG-342
Another quantity which arises in the case of two variables X and Y is the
covariance.
Definition 4. Let X and Y be random variables with probability distribution
f (x, y). The covariance of X and Y is
XX
σXY ≡ Cov(X, Y ) = E[(X − µX )(Y − µY )] = (x − µX )(y − µY )f (x, y)
x y
Example 3.5: The fraction X of male runners and the fraction Y of female runners
who complete marathon race is described by the joint density function
(
8xy, 0 ≤ x ≤ 1, 0 ≤ y ≤ x
f (x, y) =
0, elsewhere
Exercise 3.1:
Murad Ridwan, 5 of 6
Dep. of Electrical & Computer Engineering
AiOT, Addis Ababa University.
Jul 2010.
Class Notes on
3.3. COVARIANCE Applied Probability and Statistics ECEG-342
3. Let X represent the number that occurs when a green die is tossed and Y
the number that occurs when a red die is tossed. Find the variance of the
random variable
(a) 2X − Y
(b) X + 3Y − 5
175 175
Answer : (a) 12 (b) 6
1 29 13 17 17 32 230 55 20
Answer : 42 , 21 , 7 , 7 , 7 , 7 , 441 , 491 , − 147
Murad Ridwan, 6 of 6
Dep. of Electrical & Computer Engineering
AiOT, Addis Ababa University.
Jul 2010.