Upload
shanky-jain
View
231
Download
0
Embed Size (px)
Citation preview
8/3/2019 Probability Distribution MP
http://slidepdf.com/reader/full/probability-distribution-mp 1/25
8/3/2019 Probability Distribution MP
http://slidepdf.com/reader/full/probability-distribution-mp 2/25
1/9/2012
A Probability distribution is the same as Frequency Distribution
A probability indicates how probable each outcome is.
It can also be presented in graphical form e.g histogram
We can use summary measures like mean & std. Dev. To
Summarize probability distributions
We need to identify the set of outcomes to construct a
probability distribution.
8/3/2019 Probability Distribution MP
http://slidepdf.com/reader/full/probability-distribution-mp 3/25
1/9/2012
Probability Distribution
No of
women(x)
P(x)
0 0.125
1 0.375
2 0.375
3 0.125
8/3/2019 Probability Distribution MP
http://slidepdf.com/reader/full/probability-distribution-mp 4/25
Tohelp protectyour privacy, PowerPointprevented thisexternalpicturefrom being automatically downloaded.To download and display thispicture,click Optionsin the MessageBar, and then click Enableexternalcontent.
1/9/2012
Random VariableThe outcome of an experiment need not be a number, for
example, the outcome when a coin is tossed can be 'heads' or
'tails'. However, we often want to represent outcomes as
numbers. A random variable is a function that associates a uniquenumerical value with every outcome of an experiment. The value
of the random variable will vary from trial to trial as the experiment
is repeated.
There are two types of random variable - discrete and continuous.
A random variable has either an associated probability distribution
(discrete random variable) or probability density function
(continuous random variable).
8/3/2019 Probability Distribution MP
http://slidepdf.com/reader/full/probability-distribution-mp 5/25
1/9/2012
Expected ValueThe expected value (or population mean) of a random variable indicates its
average or central value. It is a useful summary value (a number) of the
variable's distribution.
Stating the expected value gives a general impression of the behaviour of
some random variable without giving full details of its probability distribution(if it is discrete) or its probability density function (if it is continuous).
Two random variables with the same expected value can have very different
distributions. There are other useful descriptive measures which affect the
shape of the distribution, for example variance.
The expected value of a random variable X is symbolised by E(X) or µ.
If X is a discrete random variable with possible values x1, x2, x3, ..., xn, and
p(xi) denotes P(X = xi), then the expected value of X is defined by:
8/3/2019 Probability Distribution MP
http://slidepdf.com/reader/full/probability-distribution-mp 6/25
1/9/2012
Discrete Random Variable A discrete random variable is one which
may take on only a countable number of
distinct values such as 0, 1, 2, 3, 4, ...
Discrete random variables are usually(but not necessarily) counts. If a random
variable can take only a finite number of
distinct values, then it must be discrete.
Examples of discrete random variables
include the number of children in a
family, the Friday night attendance at acinema, the number of patients in a
doctor's surgery, the number of defective
light bulbs in a box of ten.
Continuous Random
Variable A continuous random variable is
one which takes an infinite
number of possible values.Continuous random variables
are usually measurements.
Examples include height,
weight, the amount of sugar in
an orange, the time required to
run a mile.
8/3/2019 Probability Distribution MP
http://slidepdf.com/reader/full/probability-distribution-mp 7/25
1/9/2012
Cumulative Distribution Function All random variables (discrete and continuous) have a cumulative
distribution function. It is a function giving the probability that the random
variable X is less than or equal to x, for every value x.
Formally, the cumulative distribution function F(x) is defined to be:
for
Example
Discrete case : Suppose a random variable X has the
following probability distribution p(x )
xi 0 1 2 3 4 5
p(xi) 1/32 5/32 10/32 10/32 5/32 1/32Continued..
8/3/2019 Probability Distribution MP
http://slidepdf.com/reader/full/probability-distribution-mp 8/25
1/9/2012
The cumulative distribution function F(x) is then:
xi 0 1 2 3 4 5
F(xi) 1/32 6/32 16/32 26/32 31/32 32/32
F(x) does not change at intermediate values.
For example:
F(1.3) = F(1) = 6/32
F(2.86) = F(2) = 16/32
8/3/2019 Probability Distribution MP
http://slidepdf.com/reader/full/probability-distribution-mp 9/25
1/9/2012
Binomial Distribution
Binomial distributions model (some) discrete random variables.
Typically, a binomial random variable is the number of successes in
a series of trials, for example, the number of 'heads' occurring when
a coin is tossed 50 times. A discrete random variable X is said to follow a Binomial distribution
with parameters n and p, written X ~ Bi(n,p) or X ~ B(n,p), if it has
probability distribution where
x = 0, 1, 2, ......., n
n = 1, 2, 3, .......
p = success probability; 0 < p < 1
8/3/2019 Probability Distribution MP
http://slidepdf.com/reader/full/probability-distribution-mp 10/25
1/9/2012
The trials must meet the following requirements:
a. The total number of trials is fixed in advance;
b. There are just two outcomes of each trial; success and failure;
c. The outcomes of all the trials are statistically independent;
d. All the trials have the same probability of success.
The Binomial distribution has expected value E(X) = np and
varianceV(X) = np(1-p).
Example
8/3/2019 Probability Distribution MP
http://slidepdf.com/reader/full/probability-distribution-mp 11/25
1/9/2012
Poisson Distribution
Poisson distributions model (some) discrete random variables
Typically, a Poisson random variable is a count of the number of
events that occur in a certain time interval or spatial area. For
example, the number of cars passing a fixed point in a 5 minute
interval, or the number of calls received by a switchboard during agiven period of time.
A discrete random variable X is said to follow a Poisson distribution
with parameter m, written X ~ Po(m), if it has probability distribution
where
x = 0, 1, 2, ..., n
m > 0.
8/3/2019 Probability Distribution MP
http://slidepdf.com/reader/full/probability-distribution-mp 12/25
1/9/2012
The following requirements must be met:
a.The length of the observation period is fixed in advance;
b.The events occur at a constant average rate;
c. The number of events occurring in disjoint intervals arestatistically independent
The Poisson distribution has expected value
E(X) = m and variance V(X) = m; i.e. E(X) =
V(X) = m.
8/3/2019 Probability Distribution MP
http://slidepdf.com/reader/full/probability-distribution-mp 13/25
1/9/2012
The Poisson distribution can sometimes be used to
approximate the Binomial distribution with parameters n and
p. When the number of observations n is large, and the
success probability p is small, the Bi(n,p) distribution
approaches the Poisson distribution with the parameter given
by m = np. This is useful since the computations involved incalculating binomial probabilities are greatly reduced
Formula
P(x) = (np)x e-np
------------------x!
8/3/2019 Probability Distribution MP
http://slidepdf.com/reader/full/probability-distribution-mp 14/25
1/9/2012
Normal DistributionNormal distributions model (some) continuous random variable
strictly, a Normal random variable should be capable of assuming any value on the real line, though this requirement is
often waived in practice. For example, height at a given age for
a given gender in a given racial group is adequately described
by a Normal random variable even though heights must be
positive.
8/3/2019 Probability Distribution MP
http://slidepdf.com/reader/full/probability-distribution-mp 15/25
1/9/2012
NORMAL DISTRIBUTION
HistoryThe normal curve was developed mathematically in 1733 by DeMoivre as an
approximation to the binomial distribution. His paper was not discovered until
1924 by Karl Pearson. Laplace used the normal curve in 1783 to describe the
d ist r ibuti on of error s. Subsequently, Gauss used the normal curve to analyze
astronomical data in 1809. The normal curve is often called the Gaussian
distribution. The term bell-shaped curve is often used in everyday usage.
8/3/2019 Probability Distribution MP
http://slidepdf.com/reader/full/probability-distribution-mp 16/25
1/9/2012
Characterstics of Normal Distribution
1.It is unimodal that is it has a single peak.
2.The mean lies at the centre of the normal curve.
3.It is symmetrical thus the median &the mode also
lie at the centre of the curve.
4.The two tails extend indefinitely without touching
the horizontal axis.It is asymptotic
8/3/2019 Probability Distribution MP
http://slidepdf.com/reader/full/probability-distribution-mp 17/25
1/9/2012
The normal distribution has about 68% of the observations
lying within one standard deviation of the mean, 95% within
two standard deviations and 99.7% within 3 standard
deviations. This allows for a simple description of where most
values are to be found. The standard normal distribution has amean of 0 and a variance of 1 and is shown below.
8/3/2019 Probability Distribution MP
http://slidepdf.com/reader/full/probability-distribution-mp 18/25
1/9/2012
8/3/2019 Probability Distribution MP
http://slidepdf.com/reader/full/probability-distribution-mp 19/25
1/9/2012
Standard normal distribution-mean =o,std.dev =1
8/3/2019 Probability Distribution MP
http://slidepdf.com/reader/full/probability-distribution-mp 20/25
1/9/2012
Many distributions arising in practice can be approximated by a Normal
distribution. Other random variables may be transformed to normality.
The simplest case of the normal distribution, known as the Standard Normal
Distribution, has expected value zero and variance one. This is written as N(0,1).
Examples
8/3/2019 Probability Distribution MP
http://slidepdf.com/reader/full/probability-distribution-mp 21/25
1/9/2012
Example 1.According to HDFC bank the time taken by its customer
at the ATM is Normally distributed with a mean of18 sec.
&a std.dev of3 sec. What is the probability that a randomly
selected customer takes less than13 seconds.?
2.less than 12sec.3.less than15 sec.
4.less than18 sec.
5.less than 21sec.
x P(x)12. 0.0228
13. 0.1587
18. 0.5000
21. 0.8413
8/3/2019 Probability Distribution MP
http://slidepdf.com/reader/full/probability-distribution-mp 22/25
1/9/2012
The Z score
x -Q
Z= -------
W
Normally distributed random variables may take many different
Units of measure like inches, rupees etc so we standardise as
Z scores.
8/3/2019 Probability Distribution MP
http://slidepdf.com/reader/full/probability-distribution-mp 23/25
1/9/2012
The Normal distribution as an approximation of the Binomial dist
.Example1.Toss a coin 10 times and if we want to know the
probability of getting a head 5/6/7/8 times?
mean =np 10x0.5 = 1 std dev =sq root of npq.=1.58
compute the value for 5
+ &- a ½ as a continuity correction factor and are used for
accuracy
At x =4.5 ,
x - Q 4.5 - 5
z = ----------- = ---------- =- 0.32
W 1.58
At x =8.5, z = 2.21
8/3/2019 Probability Distribution MP
http://slidepdf.com/reader/full/probability-distribution-mp 24/25
1/9/2012
Example - Distribution of blood pressure
Distribution of blood pressure can be approximated as a
normal distribution with mean 85 mm. and standard deviation20 mm. A histogram of 1,000 observations and the normal
curve is shown below.
8/3/2019 Probability Distribution MP
http://slidepdf.com/reader/full/probability-distribution-mp 25/25
1/9/2012
THANK YOU