# 2. Probability Function

## c. interpret a cumulative distribution function;

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

How do you read the probability function: `P(X = x)`

Probability that a random variable `X`

(big X) takes on the value `x`

(little x)

What are the two key properties of a probability function?
1. `0 <= P(X=x) <= 1`

- probability is always a number between 0 and 1
2. `∑P(X=x) = 1`

- The sum of exhaustive probabilities must equal 1

Is the following example a valid probability function?
`p(x) = x/6 for X = 1, 2, 3`

for all other values `p(x) = 0`

Yes, this is a valid probability function because it satisfies both properties. I.e.:
1. `p(1) = 1/6, p(2) = 2/6, p(3) = 3/6`

- all probabilities are between 0 and 1
2. `1/6 + 2/6 + 3/6 = 6/6 = 1`

- the exhaustive values sum to 1

Is the following example a valid probability function?
`p(x) = (2x - 3)/16 for X = 1, 2, 3, 4`

for all other values `p(x) = 0`

No, this is not a valid probability function because it fails the first test. I.e.:
`p(1) = -1/16`

- a probability can’t be negative

What is the notation for the probability density function for continuous random variables?
`f(x)`

The probability for a discrete set of variables will always add up to one, and can be displayed in a table or graph. How is the probability of a continuous random variable be measured and displayed?
The continuous random variable will never have an exact value b/c there are unlimited number of values in the range. Instead, you have to measure the *area under the graph* of a range of the values. E.g., instead of saying `p(6)`

you would want the percent of the area for values between `P(5.99 <= X <= 6.01)`

. The area under the graph will always add up to 1.

##### Source:

CFA

##### Graph:

- 115.020.40 Reading 9 - Common Probability Distributions to 115.020.40.02 Reading 9 - 2. Probability Function