Discrete random variables

Question 1

What is a discrete random variable?

Answer 1

A discrete random variable transforms a sample space into a more tangible sample space whose events are more directly related to what you are more interested in.

Question 2

Give a mathematical definition of a DRV.

Answer 2

Let Ω be a sample space. A discrete random variable is a function X: Ω→R that takes on a finite number of variables or an infinite number of variables.

Question 3

How do you use a DRV?

Answer 3

One has to calculate the probabilities of X (probability distribution of X) to describe how the probability mass is distributed over the possible values of X.

Question 4

What is the deal with the probability mass function of X?

Answer 4

Once we introduce a discrete random variable the sample space is no longer relevant. It is sufficient to list all possible values of X and their corresponding probabilities. All of this is information is contained in the probability mass function of X.

Question 5

Give the mathematical definition of the probability mass function.

Answer 5

The probability mass function p of a discrete random variable X is the function p: R→[0,1], defined by:

p(a)=P(X=a) for -inf<a></a>

Question 6

What is the deal with cumulative distribution functions?

Answer 6

The probability mass function cannot specify the so-called continuous random variables. However, the cumulative distribution function of a random variable X allows us to treat discrete and continuous random variables the same way.

Question 7

Give the mathematical definition of the cumulative distribution function.

Answer 7

The distribution function F of a random variable X is the function F: R→[0,1], defined by:

F(a)=P(X≤a) for -inf<a></a>

Question 8

How can you get all the probabilistic information of a random variable?

Answer 8

You can get all of the probabilistic information of X with the probability mass function an the cumulative distribution function. You can determine the probability distribution of X with either of them. Furthermore you can express each of the function terms of the other function.

p(ai)>0 P(a1)+p(a2)+….=1
F(a)=Sum(P(ai))

Question 9

What are the three properties of the distribution function F of a random variable?

Answer 9

1. For a≤b→F(a)≤F(b). This property is derived from the
   fact that a≤b implies the the event {X≤a} is contained
   in the event {X≤b}.
2. Since F(a) is a probability, the value is always
   between 0 and 1.
3. F is right continuous.

Question 10

What values can be taken by random variables?

Answer 10

Random variables are based on the outcome of a process. It does not necessarily have to be the direct outcome, it can be the outcome of a function which uses the outcome of a process as input.

Question 11

What characterizes discrete random variables?

Answer 11

Discrete random variables have distinct and separate values.