Answer:
(a) Name: Multinomial distribution
Parameters:

(b) Range: 
(c) Name: Binomial distribution
Parameters:






Step-by-step explanation:
Given




High Slabs
Medium Slabs
Low Slabs
Solving (a): Names and values of joint pdf of X, Y and Z
Given that:
Number of voids considered as high slabs
Number of voids considered as medium slabs
Number of voids considered as low slabs
Since the variables are more than 2 (2 means binomial), then the name is multinomial distribution
The parameters are:

And the mass function is:

Solving (b): The range of the joint pdf of X, Y and Z
Given that:

The number of voids (x, y and z) cannot be negative and they must be integers; So:


Hence, the range is:

Solving (c): Names and values of marginal pdf of X
We have the following parameters attributed to X:
and 
Hence, the name is: Binomial distribution
Solving (d): E(x) and Var(x)
In (c), we have:
and 








In (b), we have: 
However, the given values of x in this question implies that:


Hence:


This question implies that:

Because
--- for x
In (e), we have:

So:


In (a), we have:

So:



Expand



Using a calculator, we have:

So:



This implies that:

In (c), we established that X is a binomial distribution with the following parameters:

Such that:

So:







So:




Y has the following parameters
and 

