Analog Signals and Systems - Digital Signal Processing: An Introduction with MATLAB and Applications - page 37

p ð A j B Þ¼ p ð A \ B Þ

p ð B Þ :

1.3.2.5 Independent Events

The events A and B are independent when p(A|B) = p(A). Hence, using the above
formula: p(A \ B) = p(A) p(B).

Example 1 A box contains 20 cubes, 15 of them red and 5 of them blue. Two
cubes are to be selected randomly (without replacement). What is the probability
that the 1st is red and the 2nd is blue?

Solution: Let R = {1st cube is red}; B = {2nd cube is blue}.

p ð R Þ¼ 15

20 ¼ 3 4 ; p ð B j R Þ¼ 5

19

p ð B \ R Þ¼ p ð R Þ p ð B j R Þ¼ 3

4 5 19 ¼ 15

76 :

Example 2 Two coins are tossed. What is the probability that the first one is a
head and the second is a tail?

Solution: p ð H \ T Þ¼ p ð H Þ p ð T Þ¼ 2 2 ¼ 4 (because they are independent
events). If one had asked for the probability that a head and a tail was obtained
from two consecutive coin tosses (without specifying the order in which they were
obtained), the answer would be 2(1/4) = 1/2.

1.3.2.6 Probability Density Function

The pdf of a random variable X, p X (X), is a non-negative function (with a total area
of 1) that shows how the values of X would be distributed if a large number of
experiments (trials) were conducted. The constraints on the pdf are expressed
mathematically as:

p X ð X Þ 0 ; Z

1

p X ð x Þ dx ¼ 1

1

p X ð x 1 X x 2 Þ¼ Z

x 2

p X ð x Þ dx :

x 1

Note that x, x 1 , and x 2 are values attained by the random variable X.

1.3.2.7 Statistical Mean

The statistical mean (or expected value) of a random variable X is defined as