Question

Let (X1, X2, X3, X4) be Multinomial(n, 4, 1/6, 1/3, 1/8, 3/8). Derive the joint mass function of the pair (X3, X4). You should be able to do this with almost no computation.

Answer 1

Answer:

The random variables in this case are discrete since they have a Multinomial distribution.

The probability mass function for a discrete random variable X is given by:

$P(X=x_{i} )$

Where are $x_{i}$ are possible values of X.

The joint probability mass function of two discrete random variables X and Y is defined as

P(x,y) =P(X=x,Y=y).

It follows that, The joint probability mass function of $X_{3} , X_{4}$ is :

$P(X_{3}, X_{4} ) = P( X_{3} = x_{3}, X_{4} = x_{4} ) =\frac{1}{8} +\frac{3}{8} =\frac{1}{2}$