### GIGAMIND

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115 CFA
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115.020.40.02 Reading 9 - 2. Probability Function

# 2. Probability 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.

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