Download Outline Continuous random variable revisited Probability Density

Probability and Statistics
LECTURE 5
CONTINUOUS PROBABILITY
DISTRIBUTIONS AND
SAMPLING METHODS
Outline
1. Probability density function
2. Normal distribution
3. Sampling methods
Adapted from http://www.prenhall.com/mcclave
3-1
3-2
Continuous random
variable revisited
Questions:



What is a continuous random variable?
Give some examples.
Can we represent probability distributions for
continuous random variables the same way
as for discrete random variables?
3-3
Probability Density Function
1. f(x)  0 for all values of x
2. The area under the
probability density
function f(x) over all
values of the random
variable X within its
range, is equal to 1.0
3-4
Probability density
function
f(x)
x
Measuring probability for
continuous random variables
d
Probability Is Area
Under Curve!
P(c  x  d )  c f ( x ) dx
f(x)
c
3-5
d
•
•
© 1984-1994 T/Maker Co.
Specific distribution having a characteristic
bell-shaped form.
The most important continuous probability
distribution.
Why normal distributions are important?
3-7
Suppose X is a continuous random
variable, does it make any difference if we
write: P(a  X  b) or P(a  X  b) or P(a 
X  b) or P(a  X  b)?
X
Normal Distribution
•
Small discussion
3-6
An example (hypothetical)
Suppose we gather data about mathscores
of all10th graders in country A.
Plot a histogram: see next slide.
Histogram and smooth curve.
3-8
Histogram
3-9
3 - 10
Normal Distribution
•
•
•
•
Normal probability density
function
Many normal distributions
Bell-shaped and symmetric
Mean = median = mode
A normal distribution is completely
determined by µ and 
Notation:
3 - 11
3 - 12
Standardize the
Normal Distribution
Finding probabilities of
normal random variables
•
•
•
Probability of a single value
Probability of X within an interval
What if we try to table the normal
distribution probabilities?
3 - 13
Normal
Distribution
Obtaining
the Probability
Table 1 in your textbook
P(X < 13.3) = P(Z < 0.83)
3 - 15
Notation: Z ~ N(0,1)
3 - 14
Standardizing Example
Suppose X is normally distributed with
mean = 5, and standard deviation = 10,
P(X < 13.3) = ?
First convert to Z:
Standard Normal Distribution
3 - 16
Solution
P(3.8  X  5)
Exercise: P(3.8  X  5)=?
Normal
Distribution
Normal
Distribution
3 - 17
3 - 18
•
3 - 19
P(Z < 0) – P(Z < -0.12)
Shaded area exaggerated
Solution
P(X  8)
Exercise: P(X  8)=?
Normal
Distribution
P(3.8  X  5) = P(-0.12 Z  0) =
We have P(X  8) = P(Z  0.30) = 1 
P(Z < 0.30) = 1- 0.6179 = 0.3821
3 - 20
Self-study: Finding Z Values
for Known Probabilities
What is z* given
P(Z < z*) = .7704?
Self-study: Finding X Values
for Known Probabilities
Example: part b of the exercise in the
next slide
What is z* given
P(Z > z*) = 0.1935
3 - 21
3 - 22
Solution
Problem
You work in Quality Control for GE.
Light bulb lifetime has a normal
distribution with
= 2000 hours & = 200 hours.
a. What’s the probability that a
randomly selected bulb will last
between 2000 & 2400 hours?
b. (Self-Study) Find the lifetime
range of the 5% most durable
light bulbs.
3 - 23
3 - 24
Sampling
Sampling with/without
replacement
Some terms:
•
•
Census:
Sampling: selecting a sample from population
Sampling with replacement:
Why sampling is necessary?


Cost
Practicality
Sampling without replacement:
Statistical inference will allow us to draw
conclusions for population based on sample
data
3 - 25
3 - 26
Sampling methods
Probability sampling methods


Probability sample: a sample selected in
such a way that each item or person in the
population has a known (nonzero)
likelihood of being included in the sample.
E.g. Simple random sampling, systematic
random sampling, and stratified random
sampling.
3 - 27
Sampling methods
Nonprobability sampling methods



Not all items or people have a chance of
being included in the sample.
E.g. online survey, surveying customers
by going around a shopping mall and
asking customers to answer a
questionnaire.
Results may be biased.
3 - 28
Some Important Probability
Sampling Methods
Let’s now discuss two of the most
important probability sampling
methods: simple random sampling
and stratified random sampling.
3 - 29
Simple random sampling
Example (cont.): we first have to obtain a
list of population members. Number the list
from 1 to N (the population size).
Then, use a computer software to select a
random sample.
This type of sampling will be demonstrated
in R labs.
Simple random sampling
Selecting a sample in such a way that
every possible sample of the same size
is equally likely to be chosen.
Example: Suppose our lecture class is
the population. We are going to select a
simple random sample (SRS) of 20
students from the population.
3 - 30
Stratified random sampling
To select a stratified random sample,
first divide the population into mutually
exclusive groups of similar individuals
called strata. Then choose a separate
SRS from each stratum and combine
these SRSs to form the full sample.
Example:
3 - 31
3 - 32
Other types of sampling
Please refer to your textbook for other
type of sampling methods
3 - 33
One student volunteers to summarize
the main points of the lecture.
3 - 34
Conclusion
1. Probability density function
2. Normal distribution
3. Sampling methods
3 - 35
Lecture summary