# 3. Cumulative Distribution Function

## c. interpret a cumulative distribution function;

## d. calculate and interpret probabilities for a random variable, given its cumulative distribution function;

What kind of function do you want if you're looking to get the probability that a random variable X is less than or equal to a particular value?

You want the cumulative distribution function (CDF), `P(X<=x)`

How do you get the cumulative distribution for any set of values in a probability function?

The cumulative distribution sums all values less than or equal to `x`

. If you have a probability function and want the CDF for `P(X<=.5)`

you would sum the probability of all values for `X<=5`

.

What are the two defining characteristics of the cumulative distribution function (CDF)?

1. The cumulative distribution function lies between 0 and 1 for any `x: 0 ≤ F(x) ≤ 1`

2. As we increase `x`

, the cdf either increases or remains constant.

In general, given the cumulative distribution function, what is the math formula for the probabilities of a random variable?

$$P(X = x_n) = F(X_n) - F(X_{n - 1})$$

What is a cumulative frequency distribution and how is it displayed?

- A cumulative frequency distribution is a plot of the number of observations falling in or below an interval.

- It can show either the actual frequencies at or below each interval or the percentage of the scores at or below each interval.

- The plot can be a histogram as or a polygon.

##### Source:

- CFA