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Probability · Lecture 30 of 76 · 5:24

Normal Probability Calculation

Study guide

What this lecture covers

This short recitation problem drills the mechanics of computing probabilities for normal random variables using a standard normal table. It builds directly on the normal distribution and standardization technique introduced earlier in the course.

You'll practice reading probabilities from a standard normal table, handling negative values by using symmetry, standardizing a general normal random variable into standard normal form, and combining these steps to compute the probability of falling in an interval.

Key ideas

Reading the standard normal table: for X standard normal, P(X <= x) is looked up directly as phi(x) for non-negative x.
Symmetry for negative values: since the standard normal is symmetric around 0, P(X <= -a) = 1 - phi(a) for positive a.
Linear functions of normal variables are normal: for Y normal, (Y - mean)/(shift or scale) combinations remain normal, with mean and variance found using linearity of expectation and the scaling rule for variance.
Standardizing any normal variable: subtracting the mean and dividing by the standard deviation turns any normal random variable into a standard normal.
Computing interval probabilities: P(a <= Y <= b) is found by standardizing the bounds and subtracting two table lookups, phi of the upper standardized bound minus phi of the lower one.

Before you watch

Know the normal distribution and how to standardize a normal random variable.
Be comfortable reading values from a standard normal (phi) table.
This drill assumes you've seen the lecture introducing continuous random variables and the normal distribution.

Check your understanding

Why can't negative values be looked up directly on a standard normal table?
Why does subtracting the mean and dividing by the standard deviation always produce a standard normal variable?
How would you compute P(Y > 3) for a general normal Y using this method?
Why does P(X <= 0) = 0.5 for any standard normal random variable?

Vocabulary

standard normal table (phrase): A printed table of probabilities for the standard normal distribution.
We look up the probability using the standard normal table.
mechanics (noun): The step-by-step technical details of how something is done.
This drill practices the mechanics of using the normal table.
symmetry (noun): A property where something looks the same on both sides of a center.
We use the symmetry of the normal curve for negative values.
standardize (verb): To rescale a variable so it matches a fixed, standard pattern.
We standardize the normal variable before using the table.
phi (noun): The common symbol used for the standard normal cumulative distribution function.
Phi of x gives the probability that a standard normal is below x.
interval (noun): A range of values between two numbers.
We compute the probability that Y falls in an interval.
lookup (noun): The act of finding a value in a table or reference.
The final answer needs two separate table lookups.
shift (noun): A change that moves a value up or down without changing its shape.
Subtracting the mean is a shift of the distribution.
scale (verb): To multiply or divide a value to change its size.
We scale by the standard deviation to standardize the variable.
normal distribution (phrase): A common bell-shaped probability distribution symmetric around its mean.
The problem drills probability calculations for the normal distribution.
cumulative distribution function (CDF) (phrase): A function giving the probability that a variable is less than or equal to a value.
Phi is the CDF of the standard normal distribution.
mean (noun): The average value of a distribution.
Standardizing subtracts the mean from the variable.
distribution (noun): A description of how likely each possible value of a variable is.
The normal distribution is symmetric around its center.
upper bound (noun): The highest value in a range being considered.
The interval probability uses phi of the upper standardized bound.
lower bound (noun): The lowest value in a range being considered.
The interval probability subtracts phi of the lower standardized bound.
subtract (verb): To take one number away from another.
We subtract the mean before dividing by the standard deviation.
negative (adjective): Less than zero.
Negative values require symmetry to read from the table.
positive (adjective): Greater than zero.
The table gives probabilities directly for positive values.
drill (noun): A short exercise used to practice a specific skill repeatedly.
This recitation is a drill on normal probability calculations.
practice (verb): To repeat an activity in order to improve at it.
You'll practice reading probabilities from a standard normal table.
combine (verb): To bring separate steps or parts together.
The final answer combines two separate table lookups.
directly (adverb): Without any extra steps in between.
Non-negative values can be looked up directly on the table.
general (adjective): Applying broadly, not limited to one special case.
The method standardizes any general normal random variable.
rescale (verb): To change a variable's size or spread while keeping its overall shape.
Standardizing rescales a normal variable to have mean 0 and variance 1.
look up (phrasal verb): To find information by searching in a table or reference source.
You look up the standardized value in the normal table.
assume (verb): To accept something as true without proof, for the sake of an argument.
The drill assumes you already know how to standardize a variable.

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: Jagdish Ramakrishnan

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