# Discrete Random Variables

Contents

## Summary

• If X is a discrete random variable, the function given by f (x) = P(X = x)
• The sum of all the probability will always equal to one.
• The cumulative distribution function, F(x) for a random variable X is defined as the probability of certain events that exist.
• Cumulative probabilities provide, for each value x, the probability of a result less than or equal to X, $P\left[ X\quad \le \quad x \right]$.

Discrete random variable are random variable that can take on distinct and separate variable.

A random variable is a function defined on the sample space. However, a Discrete random variables is a variable that can only take a finite or countable number of values, and have a positive probability of taking each one.

If X is a discrete random variable, the function given by:

f (x) = P(X = x)

For each value of x within the range of X is called the probability distribution (or probability function) of X.

A probability distribution is said to be a table of probabilities that possible outcomes of an experiment. It will look something like this.

 x -2 2 1 3.5 p(X) 0.21 0.34 0.24 0.21

The sum of all the probabilities will always equal to one.

Eg. adding the probabilities above shows this:

0.21 + 0.34 + 0.24 + 0.21 = 1

#### Probability Density Function

The cumulative distribution function, F(x) for a random variable X is defined as the probability of certain events that exist.

#### Example #1

Q. A discrete random variable X has the following probability distribution:

 x -1 0 1 4 p(X) 0.2 0.5 a 0.1

Compute each of the following quantities:

1)   a

2)   P(0)

3)   P(X > 0)

4)   P(X ≥ 0)

5)   P(X ≤ −2)

Solution:

1)   In order to find a we will add the values and put them equal to 1

0.2 + 0.5 + a +0.1 = 1
a = 0.2

2)   Next, we will find P(0) we will look directly at the table and p(0) has 0.5 probability. Thus the answer will be 0.5.

P(0) = 0.5

3)   For p(X > 0), since its only greater than 0 we won’t include 0, but will add the probabilities of P(X = 1) + P(X = 4). So, we will add ”a” which we calculated above to be 0.2 and 0.1 to get

0.2 + 0.1 = 0.3
P(X > 0) = 0.3

4)   For P(X ≥ 0) while calculating this, we will include 0 as it says greater than equal to 0. Thus, we get:

P(X ≥ 0) = P(X = 0) + P(X = 1) + P(X = 4)
= 0.5 + 0.2 + 0.1
= 0.8

5)   For P(X ≤ −2), since there is no value as -2 thus the answer will be 0.

#### Cumulative Distribution Function

The Cumulative Distribution Function (also written as CDF) is the cumulation of the probability of all the outcomes upto a given value. Assume we have a random variable X. Cumulative probabilities provide, for each value x, the probability of a result less than or equal to X. This can be represented as:

$P\left[ X\quad \le \quad x \right]$

#### Example #2

 x 1 2 3 4 f(x) 0.1 0.2 0.3 0.4

1)   P( X ≤ 1 )

2)   P( X ≤ 2 )

3)   P( X ≤ 3 )

4)   P( X < 4 )

Solution:

1)   In order P( X ≤ 1 ), we can see that there isn’t a value lower than 1, so we will just take the value of 1 to get 0.1.

2)   P( X ≤ 2 ), since it is less than equal to 2. We will take the value of 1 and 2.

P( X ≤ 2 ) = P(X = 1)+ P(X = 2)
= 0.1 + 0.2
= 0.3

3)   P( X ≤ 3 ) = P( X ≤ 1 ) + P( X ≤ 2 ) + P( X ≤ 3 )

= 0.1 + 0.2 + 0.4
= 0.7

4)   P( X <4 ) = P( X ≤ 1 ) + P( X ≤ 2 ) + P( X ≤ 3 ) (since, it’s not less than equal to thus, we will just take probabilities till 3)

=0.1 + 0.2 + 0.4
=0.7