Normal Probability Calculation

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

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