(Wanted) Guru of Probability Model

[Junyung Kim]

Homework Statement

See attachment. This is about Probability model question. I do not know how this equation is valid.

Homework Equations

The Attempt at a Solution

 

[Stephen Tashi]

The is a version of Bayes theorem.
For discrete events A, B

P(A and B) = P(A given B) P(B) = P(B and A) = P(B given A) P(A)

P(A given B) P(B) = P(B given A) P(A)

So P(B given A) = P(A given B) P(B) / P(A)

In you equation, "x" represents the the event that some random variable V takes the value x. The "A" represents some other event. It could involve continuous random variables. For example, it might stand for the event that another random variable W is greater than 2. The function "f(x|A) " is the conditional density function of V given that event A happens. So we can think of f(x|A) to be "the probability that V = x given event A".

On the right hand side, the P(A|x) is the probability of the even A given that V = x.
The "f(x)" is the probability that V = x.

The denominator is a way of writing P(A), the probability of A.
The discrete analog of this term comes from a rule such as:

If B,C,D are mutually exclusive events whose union is the entire space of possibilities then P(A) = P(A and B) + P(A and C) + P(A and D)
P(A) = P(A|B) P(B) + P(A|C) P(C) + P(A|D) P(D)

The analog for continuous random variables that the discrete events B,C,D must be replaced by the set of all possible values of V. Instead of a discrete sum we have an integral. $$P(A) = \int P(A|x) f(x)$$