Download MATH 1070Q - Section 6.4: The Normal Distribution

MATH 1070Q
Section 6.4: The Normal Distribution
Myron Minn-Thu-Aye
University of Connecticut
Objectives
1
Understand what a probability distribution is.
2
Understand the standard normal distribution and its table of
probabilities.
3
Understand how to use the standard normal distribution to find out
about any normal distribution.
Probability distributions
A probability distribution is a function associated to a random variable X
that tells us the probability of X being within a certain range.
The probability of X being between two values a and b is the area under
the graph of the probability distribution over the interval on the x-axis
from a to b:
Normal distributions
A normal distribution is a probability distribution with certain properties.
Here are some examples:
A normal distribution is . . .
• Bell-shaped: values close to the mean are more likely, and the
probabilities decrease as we move outwards.
• Symmetric: values are equally likely to be above or below the mean.
The standard normal distribution
A random variable has a standard normal distribution if it is normally
distributed (it has a graph similar to those on the previous slide) and:
• the mean is µ = 0.
• the standard deviation is σ = 1.
A variable with a standard normal distribution is usually denoted Z (as
opposed to X ). The standard normal distribution is so useful that we have
a table listing many probabilities associated to it (p.390-391 in our text).
The table for the standard normal distribution
Look at the table of probabilities for the standard normal distribution.
Let’s look up the probability associated to −1.13: look down the left
column and find the row labeled by −1.1, then look across this row to the
column labeled by 0.03. The probability in the table is 0.1292.
This is telling us that P(Z ≤ −1.13) = 0.1292
We can see this graphically as:
Calculations with the standard normal distribution
Let Z be a random variable with a standard normal distribution.
(a) Find P(Z ≤ 1.23)
(b) Find P(Z ≥ 0.37)
(c) Find P(−1.44 ≤ Z ≤ −0.12)
Non-standard normal distributions
What if X is a random variable with a normal distribution, but it’s not
standard, meaning its mean is µ 6= 0 and its standard deviation is σ 6= 1?
We can transform X into a standard normal variable using the formula:
Z=
X −µ
σ
For instance, suppose X is a normally distributed random variable with
µ = 23 and σ = 4, and we want to know P(X ≤ 33).
Heights of students
Let the random variable X give the heights of students at the university.
Suppose X is normally distributed with mean 68 inches and standard
deviation 5 inches.
(a) Find P(X ≤ 74)
(b) Find P(X ≥ 77)
Numbers of jelly beans
A jelly bean company sells boxes where the number of jelly beans X in
each box is normally distributed with mean 50 and standard deviation 6.25.
(a) Find the probability that a box contains at least 55 jelly beans.
(b) Find the probability that a box contains between 40 and 48 jelly beans.
Recap
1
The area under the graph of a probability distribution tells us the
probability of a random variable lying in a certain range of values.
2
A normal distribution has a bell-shaped curve.
3
We have a table of probabilities to tell us about the standard normal
distribution, and we can transform any normal distribution into a
standard normal distribution, then use probabilities from the table.