Question

Show that the conditional distribution is a valid pdf/pmf for both discrete and continuous random variables.

State the assumptions necessary to show this. (Hint: your “proof” should not be overly technical.)

Answer 1

(a)

To show that the conditional distribution is a valid pmf for discrete random variable:

(i)

The assumptions necessary to show this :

By definition of Conditional Distribution, we have:

Let (X,y) be a discrete bivariate random vector with Joint Probability Mass Function (pmf) f(x,y) and Marginal Probability Mass Function f_X (x,y) and f_Y (y).

For any value of x, such that

,

the Conditional pmf of Y given X = x is the function of y denoted by f(y/x) and defined by:

                              (1)

For any value of y, such that

,

the Conditional pmf of X given Y = y is the function of x denoted by f(x/y) and defined by:

(ii)

To show that the conditional distribution is a valid pmf for discrete random variable:

Condition 1 is satisfied because

Using (1), we get:

f(y/x) 0 for every y since f(x,y) 0 and f_X (x) > 0

Condition 2 is satisfied because

So,

both conditions are satisfied and hence the conditional distribution is a valid pmf for discrete random variable:

(b)

To show that the conditional distribution is a valid pdf for continuous random variable:

(i)

The assumptions necessary to show this :

By definition of Conditional Distribution, we have:

Let (X,y) be a continuous bivariate random vector with Joint Probability Density Function (pdf) f(x,y) and Marginal Probability Density Function f_X (x,y) and f_Y (y).

For any value of x, such that

,

the Conditional pdf of Y given X = x is the function of y denoted by f(y/x) and defined by:

                              (2)

For any value of y, such that

,

the Conditional pdf of X given Y = y is the function of x denoted by f(x/y) and defined by:

(ii)

To show that the conditional distribution is a valid pdf for continuous random variable:

Condition 1 is satisfied because

Using (2), we get:

f(y/x) 0 for every y since f(x,y) 0 and f_X (x) > 0

Condition 2 is satisfied because

So,

both conditions are satisfied and hence the conditional distribution is a valid pdf for continuous random variable: