**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*, .

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:

#### 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