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115 CFA
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115.020.40.03 Reading 9 - 3. Cumulative Distribution Function

# 3. 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.

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