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

Marks: 10M

Year: May 2016

Total Probability Theorem:

Statement: If $B_1$,$ B_2$, ……….$B_n$ be a set of exhaustive and mutually exclusive events and A is another event associated with (or caused by) B_i, then

Proof:

The inner circle represents the event A. A can occur along with (or due to) $B_1$,$ B_2$, ……….$B_n$ that are exhaustive and mutually exclusive.

∴$AB_1$,$AB_2$,$AB_3$,$AB_4$…………………..$AB_n$ are also mutually exclusive.

∴A= $AB_1$+$AB_2$+$AB_3$+$AB_4$……+$AB_n (By Addition Theorem) ![enter image description here][3] ![enter image description here][4] (A) (Using conditional probability) P(AB)=P(A∩B)= P(B).P(B/A)=P(A).P(A/B) **Bayes’ Theorem or Theorem of Probability of causes** **Statement:** If $B_1$,$ B_2$, ……….$B_n$ be a set of exhaustive and mutually exclusive events associated with a random experiment and A is another event associated with (or caused by) $B_i$, then

i=1, 2,….,n

Proof:

We know Conditional Probability is given as: P($AB_i$ )=P(A∩$B_i$ )=P($B_i$ ).P(A/$B_i$ )=P(A).P($B_i$/A) (1)

∴P($B_i$/A)=(P($B_i$ ).P(A/$B_i$ )/P(A) (2)

Now using Total Probability Theorem we have,

From equation (2) and equation (3)

HENCE PROVED.