1853_Random Variable & Distribution.pptx

Random Variable & Distribution

Random Variables
Definition :
A random variable is function that assigns a real
number X(s) to every elements sS where S is
the sample space corresponding to a random
experiment.
i.e. X: S → R
The function X(s)=x that maps the elements of
the sample space into real numbers is called R.V.

Type of random variables
• Discrete random variable
• Continuous random variable.

• Discrete random variable:
• If X is a R. V. which can take a finite number or
countably infinite number of values, X is called
discrete random variable.
• When a R.V. is discrete, the possible values of X
may be assumed as
x1, x2, x3, ……, xn, ……….
In the finite case, the list of values terminates and
in the countably infinite case, the list goes up to
infinity.
Ex.

Probability mass function (p.m.f.)
If X is a R. V. which can take the values x1, x2, x3, ……,
xn, ………. s.t. P(X = xi) = pi, then pi
Is called probability mass function (p.m.f.) or point
probability function, provided pi (i=1, 2, …)
satisfy the following condition:
1. pi ≥ 0 for all i, and
2. ,
The collection of pairs {xi, pi}, I =1, 2, 3,….., is called
the probability distribution of the R.V. X
1
i
i
p 


Continuous random variable:
• If X is a R. V. which can take all values (i.e., infinite
number of values)in a certain interval, then x is
called a continuous RV.
• In other words a random variable is said to be
continuous when its different values cannot be put in
1-1 correspondence with a set of positive integers.
• A continuous random variable is a random variable
that can be measured to any desired degree of
accuracy. Examples of continuous random variables
are age, height, weight etc.

• Probability density function(p.d.f.):
If X is a continuous RV s.t.
f(x) dx represents the probability that X falls in
the extremely small interval .
Then f(x) is called the probability density
function of X, provided f(x) satisfies the
following conditions:
1. .f(x) ≥0 for all xЄRx, and
2. .
1 1
( )
2 2
P x dx X x dx f x dx
 
    
 
 
1 1
,
2 2
x dx x dx
 
 
 
 
( ) 1
x
R
f x dx 


• The probability for a variate value to lie in the interval dx is f(x)
dx and hence the probability for a variate value to fall in the
finite interval [a, b] is:
Important Remark.
• In case of discrete random variable the probability at a point,
i.e., P (x = c) is not zero for some fixed c.
• However, in case of continuous random variables the probability
at a point is always zero, i.e., P (x = c) = 0 for all possible values
of c. This follows directly from (*) by taking a= b = c.
( ) ( ) ..............(*)
b
a
P a X b f x dx
  

This property of continuous r.v., viz.,
P(X= c)= 0, for every c
leads us to the following important result :
i.e., in case of continuous R.V., it does matter
whether we include the end points of the
interval from a to b.
( ) ( ) ( ) ( )
P a X b P a X b P a X b P a X b
          

Mathematical Expectation
If X is a discrete R. V. with p.m.f. P(X = xi) =
pi; then mathematical expectation of X or
arithmetic mean of X, denoted as E(X), is
defined by
E(X) = pi.
The mathematical expectation is the sum of
products of the different possible values of
the random variable and the corresponding
probabilities.
If X is a continuous R. V. with p.d.f. f(x), then
mathematical expectation of X is defined by
E(X) =

• Properties of expected values:
a) E(X+Y) = E(X) + E(Y), provided all
expectation exists.
b) If X and Y are independent random
variables then E(XY) = E(X). E(Y).
c) In general, E(XY)  E(X). E(Y).

Variance:
The variance of a RV X is defined as
Var (X) = σ2
= E [(X - )2
]
Where = E(X) is the mean of RV X.
Therefore
Var(X) =
Theorem: If X is a RV and a and b are
constants then V(aX + b) = a2
V(X).

Discrete probability distribution
• In this section we shall discuss some of the
probability distributions that figure most
prominently in statistical theory and application.
• We shall also study their parameter and derivation
of some of their important characteristics.
• We shall introduce number of discrete probability
distribution, that have been successfully applied in a
wide variety of decision situations.
• The purpose of this section is to show the type of
situations in which these distributions can applied.

• This section devoted to the study of univariate
distributions like
1. Discrete uniform (rectangular) distribution
2. Binomial distribution
3. Poisson distribution
4. Geometric distribution &
5. Hypergeometric distribution
We have already defined distribution function,
mathematical expectation and variance. This
prepares us for a study of discrete theoretical
probability distribution.

Binomial distribution
• Let a random experiment be performed repeatedly and let
the occurrence of an event in a trial be called a success
and its non-occurrence a failure.
• Consider a set of n independent trials (n being finite) in
which the probability 'p' of success in any trial is constant
for each trial. Then q = 1 - p, is the probability of failure in
any trial.
• Definition.
If random variable X is said to follow binomial distribution if it
assumes only non-negative values and its 'Probability mass
function is given by
P(X = x) =

• The two independent constants n and p in the
distribution are known as the parameters of the
distribution.
• 'n' is also, sometimes known as the degree of the
binomial distribution.
• We shall use the notation
XB (n, p)
to denote that the random variable X follow the
binomial distribution with parameters n and p.

Physical conditions for Binomial Distribution.
We get the binomial distribution under the following
experimental conditions.
1) Each trial results in two mutually disjoint outcomes
termed as success and failure.
2) The number of trials ‘n' is finite.
3) The' trials are independent of each other.
4) The probability of success 'p’ is constant for-each trial.

• Mean and variance of binomial distribution
Mean = E(X) = np
Var(X) = npq

Poisson Distribution
Poisson distribution is a distribution related to probabilities of events which are
extremely rare, but which have a large number of independent opportunities
for occurrence.
Following are some instances where Poisson distribution may be successfully
employed
1) Number of deaths from a disease. Such as heart attack or cancer or due to
snake bite.
2) Number of suicides reported in a particular city.
3) The number of defective material in a packing manufactured by a good
concern.
4) Number of faulty blades in a packet of 100.
5) Number of air accidents in some unit of time.
6) Number of printing mistakes at each page of the book.
7) Number of cars passing a crossing per minute during the busy hours of a
day.

• Poisson distribution is a limiting case of the
binomial distribution under the following
conditions:
1) n, the number of trials is indefinitely large, i.e.,
n→∞.
2) p, the constant probability of success for each
trial is indefinitely small, i.e., p→0 .
3) np = λ, (say), is finite. Thus p = , q = 1 - , where λ
is a positive real number.

• Definition:
• A random variable X is said to follow a Poisson
distribution if it assumes only non-negative
values and its probability mass function is given
by
• P(X = x) =
• Here λ is known as the parameter of the
distribution.
• We shall use the notation XP (λ) to denote that
X is a Poisson variate with parameter λ.

• Mean and variance of Poisson distribution
• Mean = E(X) = λ.
• Var X = λ

Continuous probability distribution
• In this section we shall discuss some of the
probability distributions that figure most
prominently in statistical theory and applications.
• We shall also study their parameter and derivation
of some of their important characteristics.
• The purpose of this section is to show the type of
situations in which these distributions can applied.

This section devoted to the study of univariate
distributions like
1. Continuous uniform (rectangular) distribution
2. exponential distribution
3. Normal distribution
4. Chi square distribution
We have already defined distribution function,
mathematical expectation, variance and m.g.f.
This prepares us for a study of continuous
theoretical probability distribution.

Normal (Gaussian)
distribution

←------ 68.27%----→
←-----------------95.45% ------------------ →
←----------------------------- 99.73%---------------------------- →←
Area between z = ±1 is 68.27%
Area between z = ± 2 is 95.45%
Area between z = ± 3 is 99.73%

Importance of Normal Distribution.
Normal distribution plays a very important role in statistical theory
because of the following reasons :
• Most of the distributions occurring in practice, e.g., Binomial.
Poisson, Hyper geometric, distributions. etc., can be approximated
by normal distribution.
• Moreover, many of the sampling distributions. e.g., Student's ‘t' ,
Snedecor's ‘F’, Chi-square distributions, etc., tend to normality for
large samples.
• Many 'of the distributions of sample statistic (e.g., the distributions
of sample mean, sample variance, etc.) tend to normality for large
samples and as such they can best be studied with the help of the
normal curves.
• The entire theory of small sample tests, viz., t, F, X2
tests-etc. is
based on the fundamental assumption that the parent populations
from which the samples have been drawn follow normal
distribution.

• X is normally distributed and the mean of X is
12 and S.D. is 4. (a) Find out the probability of
the following:
• (a) (i). X ≥ 20, (ii) X ≤ 20 and, (iii) 0 ≤ X ≤ 12
• (b) Find x', when P (X > x') = 0·24.
• (c) Find xo
’
and x’
1 when P (xo
’
< X < x1
’
) = 0.50
and p(X > x1') = 0.25

Assume the mean heights of soldier to be 68·22
inches with a variance of 10·8 (inch) 2
. How many
soldier in a regiment of I,000.would you expect
to be over 6 feet tall? "(Given that the area
under the standard normal curve between x=0
and X=0·35 is 0.1368 and between X=0 and X=
1.15 is 0.3746).

In an intelligence test administered to 1,000
children, the average score is 42 and standard
deviation 24.
(I) Find the number of children exceeding the
score 60, and
(ii) Find the number of children with score lying
between 20 and 40.

THEOREM:
If X1 and X2 and independent then
(i) (X1 )
(ii) (X1 )

Q.
The marks obtained by the students in mathematics,
physics, and chemistry in an examination are normally
distributed with mean 52 50 and 48with standard deviation
10, 8 and 6 resp. find the probability that a student selected
at random has secured a total of
(i) 180 or above and
(ii) 135 or less.
Q.
If X and Y are independent RVs each following N(0, 3), what
is the probability that the point(X, Y) lies between the line
3X + 4Y = 5 and 3X + 4Y = 10?

1853_Random Variable & Distribution.pptx

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