Download File

Normal Distribution
Introduction
Probability Density Functions
Probability Density Functions…
• Unlike a discrete random variable which we
studied in Chapter 7, a continuous random
variable is one that can assume an
uncountable number of values.
•  We cannot list the possible values because
there is an infinite number of them.
•  Because there is an infinite number of
values, the probability of each individual value
is virtually 0.
8.3
Point Probabilities are Zero
Because there is an infinite number of values, the
probability of each individual value is virtually 0.
Thus, we can determine the probability of a range of
values only.
• E.g. with a discrete random variable like tossing a die, it is meaningful to
talk about P(X=5), say.
• In a continuous setting (e.g. with time as a random variable), the
probability the random variable of interest, say task length, takes exactly 5
minutes is infinitesimally small, hence P(X=5) = 0.
• It is meaningful to talk about P(X ≤ 5).
8.4
Probability Density Function…
•
A function f(x) is called a probability density function
(over the range a ≤ x ≤ b if it meets the following
requirements:
1) f(x) ≥ 0 for all x between a and b, and
f(x)
area=1
a
b
x
2) The total area under the curve between a and b is 1.0
8.5
8.6
Uniform Distribution…
• Consider the uniform probability distribution
(sometimes called the rectangular probability
distribution).
• It is described by the function:
f(x)
a
b
area = width x height = (b – a) x
x
=1
8.7
Example
• The amount of petrol sold daily at a service station
is uniformly distributed with a minimum of 2,000
litres and a maximum of 5,000 litres.
f(x)
2,000
5,000
x
• What is the probability that the service station
will sell at least 4,000 litres?
• Algebraically: what is P(X ≥ 4,000) ?
• P(X ≥ 4,000) = (5,000 – 4,000) x (1/3000) = .3333
8.8
Bin width 25
Bin width 5
Bin width 1
Conditions for use of the Normal
Distribution
• The data must be continuous (or we can use a
continuity correction to approximate the
Normal)
• The parameters must be established from a
large number of trials
The Normal Distribution…
•The normal distribution is the most important of all
probability distributions. The probability density function
of a normal random variable is given by:
•It looks like this:
•Bell shaped,
•Symmetrical around the mean
…
8.14
The Normal Distribution…
•Important things to note:
The normal distribution is fully defined by two parameters:
its standard deviation and mean
The normal distribution is bell shaped and
symmetrical about the mean
Unlike the range of the uniform distribution (a ≤ x ≤ b)
Normal distributions range from minus infinity to plus infinity
8.15
Standard Normal Distribution…
•A normal distribution whose mean is zero and standard
deviation is one is called the standard normal distribution.
0
1
1
•Any normal distribution can be converted to a standard
normal distribution with simple algebra. This makes
calculations much easier.
8.16
Normal Distribution…
•Increasing the mean shifts the curve to the
right…
8.17
Normal Distribution…
•Increasing the standard deviation “flattens”
the curve…
8.18
Calculating Normal Probabilities…
•Example: The time required to build a computer is
normally distributed with a mean of 50 minutes and a
standard deviation of 10 minutes:
0
•What is the probability that a computer is assembled in a
time between 45 and 60 minutes?
•Algebraically speaking, what is P(45 < X < 60) ?
8.19
Calculating Normal Probabilities…
•P(45 < X < 60) ?
…mean of 50 minutes and a
standard deviation of 10 minutes…
0
8.20
Distinguishing Features
• The mean ± 1 standard deviation covers
66.7% of the area under the curve
• The mean ± 2 standard deviation covers
95% of the area under the curve
• The mean ± 3 standard deviation covers
99.7% of the area under the curve
Tripthi M. Mathew, MD, MPH
68-95-99.7 Rule
68% of
the data
95% of the data
99.7% of the data
Are my data “normal”?
• Not all continuous random variables are
normally distributed!!
• It is important to evaluate how well the data
are approximated by a normal distribution
Are my data normally distributed?
1. Look at the histogram! Does it appear bell
shaped?
2. Compute descriptive summary measures—are
mean, median, and mode similar?
3. Do 2/3 of observations lie within 1 std dev of the
mean? Do 95% of observations lie within 2 std dev
of the mean?
Law of Large Numbers
• Rest of course will be about using data
statistics (x and s2) to estimate parameters of
random variables ( and 2)
• Law of Large Numbers: as the size of our
data sample increases, the mean x of the
observed data variable approaches the mean 
of the population
• If our sample is large enough, we can be
confident that our sample mean is a good
estimate of the population mean!
June 5, 2008
Stat 111 - Lecture 7 - Normal
Distribution
25
Points of note:
• Total area = 1
• Only have a probability from width
– For an infinite number of z scores each point has a
probability of 0 (for the single point)
• Typically negative values are not reported
– Symmetrical, therefore area below negative value
= Area above its positive value
• Always draw a sketch!