CS 3341 Lecture Notes - Lecture 10: Cumulative Distribution Function, Probability Distribution, Benjamin Disraeli
16 May 2018
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Section 4.1 Continuous Random Variables and Probability Density
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“Seeing much, suffering much, and studying much are the three pillars of learning.”
—Benjamin Disraeli
Continuous random variables
Probability mass of continuous random variables
The set of elements in the range of a random variable that have positive probability
mass is at most countable, because we can list them (as finite sets in one-to-one
correspondence to the natural numbers).
Probability Mass:
Number of 𝑥 values:
𝑃(𝑥)=1
At most one value
𝑃(𝑥)≥1
2
At most two values
𝑃(𝑥)≥1
3
At most three values
𝑃(𝑥)≥1
4
At most four values
⋮
⋮
𝑃(𝑥)≥1
𝑛
At most 𝑛 values
An interval of real numbers is uncountable, so the set of values with positive probability
mass cannot be an interval. In fact, for all continuous random variables, the probability
mass function is always equal to zero:
Section 4.1 Continuous Random Variables and Probability Density
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Cumulative Distribution Function
Without meaningful probability mass, we must use the cumulative distribution function:
𝐹(𝑥)=
The properties we have discussed are still valid:
Additionally, the CDF only “jumps” at values where 𝑃(𝑥)>0. If 𝑃(𝑥)=0 for all 𝑥, then:
Probability as Area
If the cumulative distribution function 𝐹(𝑥) is differentiable:
Section 4.1 Continuous Random Variables and Probability Density
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