Download §3.3 Normal Random Variables

The standard normal PDF
Non-standard normals
§3.3 Normal Random Variables
Tom Lewis
Fall Semester
2016
The standard normal PDF
Non-standard normals
Outline
The standard normal PDF
Non-standard normals
The standard normal PDF
Non-standard normals
Theorem
The function
2
1
x
φ(x ) = √ exp −
2
2π
is a PDF, called the standard normal PDF.
• This function was first suggested by DeMoivre in connection
with studying the PMF of the number of heads in coin tossing.
• Gauss played a role in connecting this PDF with other
phenomena.
• Notice that this function is even, that is, it is symmetric about
the y-axis.
The standard normal PDF
Non-standard normals
Definition
A random variable with PDF φ is called a standard normal random
variable and is usually denoted by Z .
The standard normal PDF
Non-standard normals
Definition
Let Z be a standard normal random variable. The CDF of Z is
usually denoted by Φ. Thus
Zz
φ(t )dt .
P (Z 6 z ) = Φ(z ) =
−∞
Some values of Φ are given in the table on page 155.
The standard normal PDF
Non-standard normals
Problem
Let Z be standard normal. Approximate each of the following using
the table on page 155.
1. P (1 < Z 6 1.5)
2. P (Z < −2)
3. P (|Z | < 1)
4. P (|Z | < 2)
5. P (|Z | < 3)
The standard normal PDF
Non-standard normals
Problem
Show that a standard normal random variable has mean 0 and
variance 1.
The standard normal PDF
Non-standard normals
Definition
A random variable Y is said to be normal if there exist real
numbers µ and σ > 0 such that
Z =
Y −µ
σ
is standard normal. In other words,
Y = σZ + µ,
an affine transformation of a standard normal random variable.
The standard normal PDF
Non-standard normals
Problem
Let Z be a standard normal random variable and let Y = σZ + µ,
σ > 0.
1. Find the density of Y .
2. Find mean of Y .
3. Find the variance of Y .
The standard normal PDF
Non-standard normals
Notation
Let µ and σ be real numbers, σ > 0. We say that
Y has an N (µ, σ2 ) distribution
provided that Y is normally distributed with mean µ and variance
σ2 (standard deviation σ).
The standard normal PDF
Theorem
The following are equivalent:
1. Y has an N (µ, σ2 ) distribution;
2. Y = σZ + µ, where Z is standard normal;
3. (Y − µ)/σ is standard normal.
Non-standard normals
The standard normal PDF
Non-standard normals
Problem
Scores on the math SAT are normally distributed with a mean of
518 and a standard deviation of 115. What fraction of the
population will score in excess of 770 on this test?
The standard normal PDF
Non-standard normals
Problem
Scores on an IQ test are normally distributed with a mean of 100
and a standard deviation of 10. Given that a person has an IQ in
the top 25%, what is the probability that their IQ will be at least
120?