Mumbai University > Electronics and Telecommunication Engineering > Sem 5 > Random Signal Analysis

Marks: 10M

Year: Dec 2015

Dependent Events (Conditional Probability)

If the occurrence of an event A is affected by the occurrence of another event B, then the events A and B are dependent. The probability of event B depending on the occurrence of event A is called conditional probability we denote it as P(B/A) and read as ‘the probability of B given A’. Similarly the probability of event A depending on the occurrence of event B is written as P(A/B).

The probability that both the dependent events A and B will occur is given by:

P(A∩B)=P(A).P(B/A) P(A)>0

=P(B).P(A/B) P(B)>0

Properties:

• Conditional probabilities P(B/A) and P(A/B) are defined if and only if P(A)≠0 and P(B)≠0 respectively.

• For P(B)>0,P(A/B)≤P(A).

• P(A/A)=1

• If A and B are independent, then P(A/B)=P(A) and P(B/A)=P(B)

∴P(A∩B)=P(A).P(B)

• The experiment with replacement leads to ‘independent events’ whereas an experiment without replacement leads to ‘dependent events’.
• If A⊂B then A∩B=A and P(A/B)=P(A∩B)/P(B)=P(A)/P(B)>P(A)
• If AℶB then A∩B=B and P(A/B)=1
• P(S/B)=P(S∩B)/P(B) =P(B)/(P(B))=1

Conditional Probability Distribution Function

Definition: Let X and Y be two random variables, discrete or continuous. The conditional distribution of the random

variable Y given X=x is

$F_Y$ (Y/X)=($F_XY$ (x,y))/($F_X$ (x) ), $F_X$ (x)>0

Similarly, the conditional distribution of the random variable X, given that Y=y is

$F_X$ (X/Y)=($F_ XY$ (x,y))/($F_Y$ (y) ), $F_Y$ (y)>0

Conditional Probability Density Function

Definition: The conditional probability density function of Y is

$f_Y$ (Y/X)=($f_XY$ (x,y))/($f_X$ (x) )

The conditional probability density function of X is

$f_x$ (X/Y)=($f_XY$ (x,y))/($f_Y$ (y) )

If X and Y are statistically independent, then

$f_x$ (X/Y)=($f_XY$ (x,y))/($f_Y$ (y) )=($f_X$ (x).$f_Y$ (y))/($f_Y$ (y) )=$f_X$ (x)

Similarly, $f_Y$ (Y/X)=$f_Y$ (y)