For a discrete random variable, its probability distribution (also called the probability distribution function) is any table, graph, or formula that gives each possible value and the probability of that value. Note : The total of all probabilities across the distribution must be 1, and each individual probability must be between 0 and 1, inclusive.

Examples:

This could be found by listing all 16 possible sequences of heads and tails for four flips, and then counting how many sequences there are for each possible number of heads.

This could be found be doing a census of a large student population.

Cumulative Probabilities

Often, we wish to know the probability that a variable is less than or equal to some value. This is called the cumulative probability because to find the answer, we simply add probabilities for all values qualifying as "less than or equal" to the specified value.

Example: Suppose we want to know the probability that the number of heads in four flips is 1 or less. The qualifying values are 0 and 1, so we add probabilities for those two possibilities.

\( P(X<2)=P(X \leq 1 )=P(X=0)+P(X=1)=(1/16)+(4/16)=5/16 \)

The cumulative distribution is a listing of all possible values along with the cumulative probability for each value

Examples:

Each cumulative probability was found by adding probabilities (in second row) up to the particular column of the table. As an example, for 2 heads, we add probabilities for 0, 1, and 2 heads to get 11/16. This is the probability the number of heads is two or less.

As an example, probability a randomly selected student has 2 or fewer tattoos = 0.985 (calculated as 0.850 + 0.120 + 0.015).