Download Continuous Random Variables - MATH 130, Elements of Statistics I

Continuous Random Variables
MATH 130, Elements of Statistics I
J. Robert Buchanan
Department of Mathematics
Fall 2015
Objectives
During this lesson we will learn to:
I
use the uniform probability distribution,
I
graph a normal curve,
I
state the properties of the normal curve,
I
explain the role of area in the normal density function.
Background
Recall:
I
A continuous random variable may take on an infinite
number of values from a range without gaps.
I
The probability that a continuous random variable lies in
particular interval depends on the shape of the distribution
for the variable.
Uniform Distribution
Suppose that the length of time a light bulb will burn (measured
in hours) is any number 0 ≤ X ≤ 1000. All intervals of equal
length are equally probable. The distribution of this random
variable is uniform.
PDF
0.0020
0.0015
0.0010
0.0005
200
400
600
800
1000
X
Equal Areas
PDF
0.0020
0.0015
0.0010
0.0005
200
400
600
800
1000
Remarks:
I
Equal areas imply equal probability X lies in those
intervals.
I
Total area under the line is 1.
X
Probability Density Function
Definition
A probability density function (PDF) is a function used to
compute probabilities of continuous random variables. The
PDF must satisfy the following two properties:
1. The total area under the graph of the PDF over all possible
values of the random variable must be 1.
2. The height of the graph must be greater than or equal to 0
for all possible values of the random variable.
The area under the graph of the PDF over an interval
represents the probability that the random variable has a value
in that interval.
Example
Find the probability that 0.5 ≤ X ≤ 1.5.
PDF
1.0
0.8
0.6
0.4
0.2
0.5
1.0
1.5
2.0
X
Solution
Using the area formula for a
triangle we have
PDF
1.0
0.8
P(0.5 ≤ X ≤ 1.5)
1
1
=
(1.5)(0.75) − (0.5)(0.25)
2
2
= 0.5625 − 0.0625
0.6
0.4
0.2
0.5
1.0
1.5
2.0
X
= 0.5
Normal Random Variables
Definition
A continuous random variable is normally distributed or has a
normal probability distribution if its probability density
function has a graph with the shape of a bell-shaped curve. The
normal probability density function is given by the formula
1
2
2
y = √ e−(x−µ) /2σ .
σ 2π
PDF
Graph of the Normal Distribution
Μ-Σ
Μ
X
Μ+Σ
Effect of Changing µ
In the figure below both normal distributions have a standard
deviation of σ = 1. The curve on the left has a mean µ = 0
while the curve on the right has mean µ = 2.
0.4
PDF
0.3
0.2
0.1
0.0
-2
0
2
X
4
Effect of Changing σ
In the figure below both normal distributions have a mean of
µ = 0. The lower curve has a standard deviation of σ = 3 and
the more pointed curve has a standard deviation of σ = 1.
0.4
PDF
0.3
0.2
0.1
0.0
-6
-4
-2
0
X
2
4
6
Properties of the Normal Distribution
1. It is symmetric about its mean µ.
2. The mean = mode = median, and thus the peak of the
curve occurs at x = µ.
3. It has inflection points at µ − σ and µ + σ.
4. The area under the curve is 1.
5. The area under the curve to the left of µ is 1/2 which is
also the area under the curve to the right of µ.
6. The PDF approaches 0 but never reaches 0.
7. The Empirical Rule applies.
Empirical Rule
34%
PDF
34%
2.35%
Μ-3Σ
13.5%
Μ-2Σ
Μ-Σ
13.5%
Μ
X
Μ+Σ
2.35%
Μ+2Σ
Μ+3Σ
Weights of M&M’s
Suppose 500 M&M’s are weighed and the data is presented in
the histogram below.
60
40
20
44
46
48
50
For this sample x = 47.79 and s = 1.46 grams.
Question: are the data normally distributed?
52
Overlay With Normal Curve
44
46
48
50
52
Standard Normal Random Variable
Definition
Suppose the random variable X is normally distributed with
mean µ and standard deviation σ, then the random variable
Z =
X −µ
σ
is normally distributed with mean µ = 0 and standard deviation
σ = 1. The random variable Z is has the standard normal
distribution.
Example
The mean salary of new hires at a certain company is $36,000
with a standard deviation of $1,500. The salaries of new hires
are normally distributed. Suppose the salary of one new hire is
X = $35, 500, what is the standardized value of this employee’s
salary?
Example
The mean salary of new hires at a certain company is $36,000
with a standard deviation of $1,500. The salaries of new hires
are normally distributed. Suppose the salary of one new hire is
X = $35, 500, what is the standardized value of this employee’s
salary?
Z =
35, 500 − 36, 000
= −0.333
1, 500
Relation Between Normal and Standard Normal
0.5
0.00025
0.4
0.00020
0.3
0.00015
0.2
0.00010
0.1
0.00005
30 000
32 000
34 000
36 000
38 000
40 000
42 000
X
The shaded areas are equal in size.
-4
-2
0
2
4
Z