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Probability · Lecture 27 of 76 · 8:44

Calculating a Cumulative Distribution Function (CDF)

Study guide

What this lecture covers

This recitation problem practices two closely linked skills for continuous random variables: finding an unknown normalizing constant in a probability density function, and integrating that density to build its cumulative distribution function (CDF). It builds on the definitions of PDF and CDF introduced earlier in the course.

You'll see how the requirement that a PDF integrates to 1 pins down an unknown constant, and how to carefully split the real line into ranges (where the density is zero, and where it isn't) to write out a complete, piecewise CDF formula.

Key ideas

A PDF must integrate to 1: setting integral of f(z) dz over the density's support equal to 1 is how you solve for an unknown constant in the density formula.
Only the support matters for integration: since the density is zero outside its given range, the normalization integral only needs to be taken over that range.
The CDF accumulates probability from the left: F(z) = integral from -infinity to z of f(y) dy represents P(Z <= z).
Keeping track of ranges is the hard part: a CDF built from a piecewise PDF needs separate formulas for where the density is zero (CDF is 0), where it's fully accumulated (CDF is 1), and the range in between (CDF is the integral of the density).
The CDF's lower and upper bounds: below the support of the density the CDF is 0, and above it the CDF is 1, since all probability mass has been accumulated.
Verifying the result: once you have gamma, you can substitute it back to fully specify the PDF before computing the CDF.

Before you watch

Know the definitions of a probability density function and a cumulative distribution function for continuous random variables.
Be comfortable with basic definite integration, including integrating polynomials.
This drill assumes you've seen the introductory continuous random variables lecture in this course.

Check your understanding

Why must you find gamma before you can compute the CDF?
Why does the CDF equal 1 for all z greater than the upper bound of the PDF's support?
How does the number of ranges needed in a CDF formula relate to the shape of the underlying PDF?
What would go wrong if you forgot to restrict the normalization integral to the PDF's support?

Vocabulary

normalizing constant (phrase): A fixed number multiplied into a formula so that its total probability equals 1.
We solve for the normalizing constant in the density formula.
unknown constant (phrase): A fixed value in a formula that has not yet been calculated.
The unknown constant is found using the rule that the density integrates to 1.
piecewise (adjective): Made up of different formulas that apply to different ranges.
The CDF is written as a piecewise function.
support (noun): The range of values where a density function is not zero.
The density is zero outside its support.
range (noun): A set of values between a lower and an upper limit.
We split the number line into separate ranges.
accumulate (verb): To gradually collect or add up over time.
The CDF accumulates probability from the left.
lower bound (phrase): The smallest value in a range.
Below the lower bound, the CDF is always 0.
upper bound (phrase): The largest value in a range.
Above the upper bound, the CDF is always 1.
substitute (verb): To put one value or expression in place of another.
We substitute the constant back into the density formula.
definite integration (phrase): Calculating the exact area under a curve between two fixed points.
Definite integration gives the exact probability over an interval.
polynomial (noun): An expression made of terms with variables raised to whole-number powers.
The density here is a simple polynomial in z.
verify (verb): To check that a result is correct.
We verify the answer by substituting it back into the original formula.
PDF (phrase): A curve that describes how likely different values of a continuous variable are.
We first find the unknown constant in the PDF.
CDF (phrase): A function giving the probability that a random variable is less than or equal to a value.
We build the full CDF by integrating the PDF.
drill (noun): A short practice exercise aimed at a specific skill.
This recitation problem is a drill on finding a CDF.
split (verb): To divide something into separate, simpler parts.
We split the real line into ranges where the density is zero or not.
density (noun): The value of a PDF at a specific point, showing how likely values are near that point.
The density is zero outside its given range.
formula (noun): A fixed rule written with symbols that gives a result from given values.
We write out a complete, piecewise CDF formula.
real line (phrase): The set of all real numbers, pictured as a straight line.
We split the real line into separate ranges for the CDF.
integrate (verb): To add up a continuous quantity over a range of values.
We integrate the density from negative infinity to z.
specify (verb): To state something exactly and in detail.
Finding gamma fully specifies the PDF before computing the CDF.
restrict (verb): To limit something to a smaller range or set.
The normalization integral must be restricted to the PDF's support.
hard part (phrase): The most difficult or error-prone step in solving a problem.
Keeping track of ranges is the hard part of building the CDF.
closely linked (phrase): Strongly connected, so that understanding one thing helps with the other.
Finding the constant and building the CDF are two closely linked skills.
zero outside (phrase): Equal to zero for all values outside a certain range.
The density is zero outside its support.

Chapters

0:00 <Untitled Chapter 1>
3:05 calculate the cdf
5:32 integrate from negative infinity to z
7:15 write out the entire cdf

From the YouTube description

MIT 6.041SC Probabilistic Systems Analysis and Applied Probability, Fall 2013
View the complete course: http://ocw.mit.edu/6-041SCF13
Instructor: Jimmy Li

License: Creative Commons BY-NC-SA
More information at http://ocw.mit.edu/terms
More courses at http://ocw.mit.edu

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