Lecture 10: The Normal Distribution
1. Continuous Random Variables
• So far all the random variables have been discrete.
• We need a different type of model (called a probability density function) for continuous random variables.
• For these models, probability corresponds to area under the
density function.
• Probability via density functions is something like proportions via histograms.
Properties of density functions
We want areas under density functions to correspond to probabilities. So
• Density functions can’t be negative.
• The total area under a density function has to be one.
• If a density function f is used to model a random variable
X, then the probability that X is between two numbers a
and b is the area under the density function between a and
b. That is
P{a ≤ X ≤ b} =
¡
¢ ¡
¢
area to left of b − area to left of a .
• For any fixed value a, P{X = a} = 0.
Exercise 6.1 page 257. Use (c) to find P{1 < X < 2}.
100p-th percentile. Let X be a continuous random variable.
Then the 100p-th percentile of the distribution is the value x
such that
P{X ≤ x} = p
In words, the area to the left of x is p and the area to the right
of x is (1 − p).
Standardized random variable: Let X be a random variable with mean µ and standard deviation σ. Then the standardized variable of X is
Z=
Exercise 6.11 (a).
X −µ
.
σ
2. The standard Normal density
There are many normal densities. But if we can learn to work
with one of them, called the standard normal density, we’ll be
able to work with all of them. The standard normal density
• Is bell-shaped
• Is symmetric about 0.
• Has almost all its area between −3 and 3.
• Has mean µ = 0 and standard deviation σ = 1.
Areas under the standard normal density: To compute
areas under the standard normal density, we will use Table 3 in
the text.
Here is a portion of the table:
Second decimal place in z
z
.00
.01
.02
.03
...
0.0 .5000 .5040 .5080
.5120
...
0.1 .5398 .5438 .5478
.5517
...
0.2 .5793 .5832 .5871
.5910
...
0.3 .6179 .6217 .6255
.6293
...
0.4 .6554 .6591 .6628
.6664
...
The numbers in the table are areas to the left of the number
z. For example:
• The area to the left of 0 is .5000.
• The area to the left of 0.03 is .5120.
• The area to the left of 0.32 is .6255.
• The area to the left of 0.41 is .6591.
Examples:
Question: What is the area under the standard normal density to the left of 2.25?
Answer: Just look up 2.25 in the table to get the answer,
.9878.
Question: What is the area under the standard normal density to the right of 2.25?
Answer: Since the total area under the density is 1, the
answer is 1 − .9878 = .0122.
Question: What is the area under the standard normal density between −1 and 2?
Answer:
• First, the area to the left of 2 is .9772.
• Second, the area to the left of −1 is .1587.
• So the answer is .9772 − .1587 = .8185.
Examples 2 & 3 see page 263 – 264.
Finding percentiles of the standard normal density
• Finding percentiles from Table 3 will be more important
than finding areas.
• Remember that since probabilities are areas,
– The 55th percentile is the number with area 0.55 to its
left.
– The 49th percentile is the number with area 0.49 to its
left.
– The 98th percentile is the number with area 0.98 to its
left.
To find percentiles:
• Find the closest area to the area you desire in the body of
Table 3. (For example, if you want the 77th percentile, find
the closest area to 0.77.)
• The percentile is the z-value corresponding to this area.
Examples: 1. What is the 93rd percentile of a standard normal distribution?
Answer: Look for 0.93 in Table 3. The closest you’ll get is
0.9306, which corresponds to z = 1.48.
2. What is the 10th percentile of the standard normal distribution?
Answer: Table 3 gives z = −1.28.