Probability Distribution MP

8/3/2019 Probability Distribution MP

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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.



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Probability Distribution

No of

women(x)

P(x)

0 0.125

1 0.375

2 0.375

3 0.125



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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).



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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:



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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.



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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..



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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



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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



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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



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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.



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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.



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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!



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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.



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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.



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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



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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.



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Standard normal distribution-mean =o,std.dev =1



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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



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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



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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.



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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



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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.



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THANK YOU

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Probability Distribution MP