High lights on probability distribution
Binomial Distribution
• It is one of the most widely used discrete
probability distributions.
• Consider dichotomous (binary) random variable
• It is based on Bernoulli trial
– When a single trial of an experiment can
result in only one of two mutually exclusive
outcomes (success or failure; dead or alive;
sick or well, male or female)
Example:
• We are interested in determining whether a newborn
infant will
survive until his/her 70th birthday
• Let Y represent the survival status of the child at age
70 years
• Y = 1 if the child survives and Y = 0 if he/she does not
• The outcomes are mutually exclusive
• Suppose that 72% of infants born survive to age 70 years
P(Y = 1) = p = 0.72
P(Y = 0) = 1 − p = 0.28
Characteristics of a Binomial Distribution
 The experiment consist of n identical trials.
 Only two possible outcomes on each trial.
 The probability of A (success), denoted by p, remains the same from trial to
trial.
 The probability of B (failure), denoted by q,
 q = 1- p.
 The trials are independent.
n and  are the parameters of the binomial distribution.
 The mean is n and the variance is n(1- )
The Poisson Distribution
• We are observing a count or number of events, rather
than a binary outcome for each subject or trial, as in the
binomial distribution .
• Applicable for counts of events over a given interval of
time, for example:
– number of patients arriving at an emergency
department in a day
– number of new cases of HIV diagnosed at a
clinic in a month
 In theory, a random variable X is a count that can
assume any integer value greater than or equal to 0
B. Continuous Probability Distributions
• A continuous random variable X can take on any value in
a specified interval or range
• With a large number of class intervals, the frequency
polygon begins to resemble a smooth curve.
• The probability distribution of X is represented by a
smooth curve called a probability density function f(x)
Distribution of serum
triglyceride
• The area under the smooth curve is equal to 1
• The area under the curve between any two points x1 and x2 is the probability that X takes a
value between x1 and x2
• Instead of assigning probabilities to specific outcomes of the random variable X, probabilities are
assigned to ranges of values
3. The Normal distribution
The ND is the most important probability distribution in
statistics
Frequently called the “Gaussian distribution”/bell-shape curve.
Variables such as blood pressure, weight, height, serum
cholesterol level, and IQ score are approximately normally
distributed
• A random variable is said to have a normal
distribution if it has a probability distribution
that is symmetric & bell-shaped
• In ND the “average "represents the true or normal
value of the measurement and deviations from this are
errors.
• Small errors would occur more frequently than large
errors.
• The ND is vital to statistical work.
• Because most estimation procedures & hypothesis tests underlie
on ND
• The concept of “probability of “X=x” in the discrete probability
distribution is replaced by the “probability density function f(x)
• The ND is also an approximating distribution to other
distributions (e.g., binomial)
1. The mean µ tells you about location – Increase µ - Location shifts right
– Decrease µ – Location shifts left
– Shape is unchanged
2. The variance σ2 tells you about narrowness or
flatness of the bell – Increase σ2 - Bell flattens. Extreme values are more likely
– Decrease σ2 - Bell narrows. Extreme values are less likely
– Location is unchanged
Properties of Normal Distribution
1. It is symmetrical about its mean, .
2. The mean, the median and mode are almost equal. It is
unimodal.
3. The total area under the curve about the x-axis is 1
square unit.
4. The curve never touches the x-axis.
5. As the value of  increases, the curve becomes more
and more flat and vice versa.
6.Perpendiculars of:
± 1SD contain about 68%;
±2 SD contain about 95%;
±3 SD contain about 99.7% of the area
under the curve.
7.The distribution is completely determined by
the parameters  &.
We have different normal distributions depending on the values of μ
and σ2.
We cannot tabulate every possible distribution
Tabulated normal probability calculations are available only for the ND with
µ = 0 and σ2=1.




Standard Normal Distribution
It is a normal distribution that has a mean equal to 0 and a SD equal
to 1, and is denoted by N(0, 1).
If a random variable X~N(,) then we can transform it to a SND
with the help of Z-transformation
These Z-scores can then be used to find the area (the probability)
under the normal curve.
We compute a standard score to transform a score from its original
units into standard deviation units.
The formula for standard scores is: Z = x - 

• Z represents the Z-score for a given x value
• A Z-score is the # of SD that a given x value is above or below the
mean
Standard normal distribution cont…
 The first standard score is a z-score for a population.
A z-score specifies the precise location of each X value within
a distribution.
The sign of the z-score (+ or -) signifies whether the score is
above the mean (positive) or below the mean (negative).
The numerical value of the z-score specifies the distance from
the mean by counting the number of standard deviation units
between X and µ.
The standard normal distribution has mean 0 and variance 1
•
Approximately 68% of the area under the standard normal curve
lies between ±1,
• about 95% between ±2, and
• about 99.7 % between ± 3
a) What is the probability that z < -1.96?
(1) Sketch a normal curve
(2) Draw a perpendicular line for z = -1.96
(3) Find the area in the table
(4) The answer is the area to the left of the line P(z < -1.96)
= 0.0250
b) What is the probability that -1.96 < z < 1.96?
The area between the values P(-1.96 < z < 1.96) =
0.9750 - 0.0250 =0.9500
c) What is the probability that z > 1.96?
• The answer is the area to the right of the line; found by
subtracting table value from 1.0000; P(z > 1.96) =1.0000 0.9750 = .0250
Applications of the Normal Distribution
• The ND is used as a model to study many
different variables.
• The ND can be used to answer probability
questions about continuous random variables.
• Following the model of the ND, a given value of
x must be converted to a z score before it can be
looked up in the z table.
Example:
• The diastolic blood pressures of males 35–44
years of age are normally distributed with µ = 80
mm Hg and σ2 = 144 mm Hg2, σ = 12 mm Hg
• Therefore, a DBP of 80+12 = 92 mm Hg lies 1 SD
above the mean
• Let individuals with BP above 95 mm Hg are
considered to be hypertensive
a. What is the probability that a randomly selected
male has a BP above 95 mm Hg?
• Approximately 10.6% of this population would be
classified as hypertensive
b. What is the probability that a randomly selected
male has a DBP above 110 mm Hg?
Z = 110 – 80 = 2.50
12
P (Z > 2.50) = 0.0062
• Approximately 0.6% of the population has a DBP
above 110 mm Hg
c. What is the probability that a randomly
selected male has a DBP below 60 mm Hg?
Z = 60 – 80 = -1.67
12
P (Z < -1.67) = 0.0475
• Approximately 4.8% of the population has a
DBP below 60 mm Hg
Sampling Distributions
Sampling distributions are important in the
understanding of statistical inference.
Definition
A Parameter :is number that can be used to describes a
population as a whole.
Statistic: is a number derived from a sample drawn
from a specific population.
In statistical practice, the value of a population
parameter is not known. A statistic is used to
estimate a parameter.
• The sampling distribution of a statistic is the distribution of all
possible values of the statistic, computed from samples of the same
size randomly drawn from the same population.
• When sampling a discrete, finite population, a sampling distribution
can be constructed.
• Note that this construction is difficult with a large population and
impossible with an infinite population
Developing a Sampling Distribution
Assume there is a population …
• Population size N=4
• Random variable, X,
is age of individuals
• Values of X: 18, 20,
22, 24 (years)
A
B
C
D
Developing a Sampling Distribution
(continued)
Summary Measures for the Population Distribution:
X

μ
P(x)
i
N
.3
18  20  22  24

 21
4
σ
 (X  μ)
i
N
.2
.1
0
2
 2.236
18
20
22
24
A
B
C
D
Uniform Distribution
x
Developing a Sampling Distribution
(continued)
Now consider all possible samples of size n=2
1st
Obs
16 Sample
Means
2nd Observation
18
20
22
24
18
18,18
18,20
18,22
18,24
20
20,18
20,20
20,22
20,24
1st 2nd Observation
Obs 18 20 22 24
18 18 19 20 21
20 19 20 21 22
22
24
22,18 22,20 22,22 22,24
16 possible samples
with
24,18 (sampling
24,20 24,22
24,24
replacement)
22 20 21 22 23
24 21 22 23 24
Developing a Sampling Distribution
(continued)
Sampling Distribution of All Sample Means
Sample Means
Distribution
16 Sample Means
1st 2nd Observation
Obs 18 20 22 24
18 18 19 20 21
20 19 20 21 22
22 20 21 22 23
24 21 22 23 24
_
P(X)
.3
.2
.1
0
18 19
20 21 22 23
(no longer uniform)
24
_
X
Developing a
Sampling Distribution
(continued)
Summary Measures of this Sampling Distribution:
μX
X


N
σX 

i
18  19  21    24

 21
16
2
(
X

μ
)
i

X
N
(18 - 21)2  (19 - 21)2    (24 - 21)2
 1.58
16
Comparing the Population with its
Sampling Distribution
Population
N=4
μ  21
σ  2.236
Sample Means Distribution
n=2
μX  21
σ X  1.58
_
P(X)
.3
P(X)
.3
.2
.2
.1
.1
0
18
20
22
24
A
B
C
D
X
0
18 19
20 21 22 23
24
_
X
Standard Error of the Mean
• Different samples of the same size from the same population will yield different sample
means
• A measure of the variability in the mean from sample to sample is given by the Standard
Error of the Mean:
• The variance of the sampling distribution is not equal to the population variance.
however, that the variance of the sampling distribution is = to the population variance
divided by the size of the sample used to obtain the sampling distribution.
• Note that the standard error of the mean decreases as the sample size increases
σX
σ

n
If the Population is Normal
If a population is normal with mean μ and standard
deviation σ, the sampling distribution of X
is also normally distributed with
μX  μ
and
σ
σX 
n
Z-value for Sampling Distribution
of the Mean
Z-value for the sampling distribution of X :
Z
where:
( X  μX )
σX
( X  μ)

σ
n
X = sample mean
μ = population mean
σ = population standard deviation
n = sample size
Sampling Distribution Properties
μx  μ
•
(i.e.
x is unbiased )
Normal Population
Distribution
μ
x
μx
x
Normal Sampling
Distribution
(has the same mean)
Sampling Distribution Properties
(continued)
As n increases,
σx
decreases
Larger
sample size
Smaller
sample size
μ
x
If the Population is not Normal
• We can apply the Central Limit Theorem:
– Even if the population is not normal,
– …sample means from the population will be approximately normal as
long as the sample size is large enough.
properties of the sampling distribution:
μx  μ
σ
σx 
n
Central Limit Theorem
As the
sample
size gets
large
enough…
n↑
the sampling
distribution
becomes
almost normal
regardless of
shape of
population
x
If the Population is not Normal
(continued)
Population Distribution
Sampling distribution
properties:
Central Tendency
μx  μ
σ
σx 
n
Variation
μ
x
Sampling Distribution
(becomes normal as n increases)
Larger
sample
size
Smaller
sample size
μx
x
How Large is Large Enough?
• For most distributions, n > 30 will give a
sampling distribution that is nearly normal
• For fairly symmetric distributions, n > 15
• For normal population distributions, the
sampling distribution of the mean is always
normally distributed
Example
• Suppose a population has mean μ = 8 and
standard deviation σ = 3. Suppose a random
sample of size n = 36 is selected.
• What is the probability that the sample mean
is between 7.8 and 8.2?
Example
(continued)
Solution:
• Even if the population is not normally
distributed, the central limit theorem can be
used (n > 30)
• … so the sampling distribution of
approximately normal
• … with mean
μx
x
is
= 8
• …and standard deviation
σ
3
σx 

 0.5
n
36
Example
(continued)
Solution (continued):


 7.8 - 8
X -μ
8.2 - 8 
P(7.8  X  8.2)  P



3
σ
3


36
n
36 

 P(-0.4  Z  0.4)  0.3108
Population
Distribution
???
?
??
?
?
?
?
?
μ8
Sampling
Distribution
Standard Normal
Distribution
Sample
.1554
+.1554
Standardize
?
X
7.8
μX  8
8.2
x
-0.4
μz  0
0.4
Z
• Up until this point, we have assumed that the values of
the parameters of a probability distribution are known.
• In the real world, the values of these population
parameters are usually not known
• Instead, we must try to say something about the way in
which a random variable is distributed using the
information contained in a sample of observations