Multiple Random Variables

PMF of multiple Random variables

We did the PMF for a single discrete random variable. So when we have multiple random variables on the same probability space, we can have many PMFs such as the Joint PMFs, Marginal PMFs, and Conditional PMFs. So first, we will define the joint PMFs of two discrete random variables. You can extend it to multiple random variables.

Let X and Y are two discrete random variables defined in the same probability space.  let the range of X and Y be and respectively. The joint PMF of X and Y denoted is a function from to [0,1] defined as

or are a set of possible values ( Range) Taken by the random variable X and Y.

Joint PMF is usually written as a table or a matrix.

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Example:  Toss a fair coin twice

Let Xi= 1 if i-th toss is heads and Xi= 0 if the i-th toss is tails, i=1,2.

As both the events are independent of each other so,

Similarly, we can compute the other combinations. The Joint PMF table will look like

For the PMFs to be valid, their value must lie between [0,1], endpoints inclusive, and the sum of all the values must give 1. If both the random variables were dependent on each other like in our earlier example, the Joint PMF table is calculated as

• (X = 0 and Y = 0) = The tens place is 0 and the two-digit number is divisible by 4. The possible outcomes are { 00, 04, 08 }. 

          Hence, P(X = 0 and Y = 0) = .

• (X = 0 and Y = 1) = The tens place is 0 and the two-digit number leaves remainder 1 if divided by 4. The possible outcomes are { 01 ,05, 09 }. Hence, P(X = 0 and Y = 1) = .
• (X = 1 and Y = 0) = The tens place is 1 and the two-digit number is divisible by 4. The possible outcomes are { 12 ,16 }. 

          Hence, P(X = 1 and Y = 0) = .

• (X = 1 and Y = 1) = The tens place is 1 and the two-digit number leaves remainder 1 if divisible by 4. The possible outcomes are { 13,17 }. Hence, P(X = 1 and Y = 1) = .

Similarly, we can sketch a joint PMF table considering all the possible combinations.

Marginal PMFs

Suppose X and Y are jointly distributed discrete random variables with joint PMF . The PMF of the individual random variables X and Y are called Marginal PMFs.

Where and are the ranges of X and Y, respectively. Consider the above joint PMF table. We can compute the Marginal PMF of X and Y for a particular value by summing up the rows or columns.

Conditional PMFs

Suppose X is a discrete random variable with range , and A is an event in the same probability space. The conditional PMF of X given A has occurred is given as

So using the formula, We calculate the conditional distribution of one random variable given another. Suppose X and Y are jointly distributed discrete random variables with Joint PMF  . The Conditional PMF of Y given X = t is defined as

Or,

Note: Range of Y|X=t can be different from Y and depend on it.

FAQs

1. Three balls are selected at random from a box containing five red, four blue, three yellow and six green coloured balls. If X, Y and Z are the number of red balls, blue balls and green balls respectively, find P(X = 1, Y = 0, Z = 2).
   P(X = 1, Y = 0, Z = 2) = P(one red ball and 2 green balls) =.
   
    
2. An experiment consists of rolling an unbiased die two times. The random variables X ∼ Uniform { 1,2,3,4,5} ,represent the number of ith roll , i = 1,2 . Calculate .
   As rolling a die two times is an independent event.
   So .
    
3. If X, Y ~ Then,
    
   
    
4. What is a Random variable?
   A Random value is the set of possible values of the random experiment.
    
5. How many types of Random variables are there?
   There are two types of Random variables; Discrete and Continuous

Key Takeaways

We looked at the Random multiple variables in great detail. I hope you find it interesting. Visit here for more information.