unit i - random variables

7/28/2019 UNIT I - Random Variables

1/12

Random Variables

Definition:A random variable X is a real - valuedmapping on a sample space.

:X

X S R

Example: Toss a Coin twice

S = (TT, TH, HT, HH)

X : Number of Heads

RX = (0, 1,2) X(TT) = 0

X(TH U HT) = 1

X(HH) = 2


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Discrete Random variable

If X is a random variable (RV) which can take a finitenumber of or countably infinite number of values, X is

called a discrete RV. When the RV is discrete the

possible values of X may be assumed as x1, x2, , xn,.

If X is a discrete RV which can takes the values x1, x2, ,xn,such that P(X=x) = pi, then pi is called the

probability function or probability mass function or point

probability function, provided pi(i = 1,2,3,) satisfy the

following conditions:( ) 0, , ( ) 1i i

i

i p for all i and ii p


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Continuous Random variable

If X is a random variable (RV) which takes all values (i.e., infinite

number of values) in an interval, then X is called a continuous RV.

For example, the length of time during which vacuum tube

installed in a circuit functions is a continuous RV.

f(x) is called the probability density function (pdf) of X, provided

f(x) satisfies the following conditions

( ) ( ) 0, , ( ) ( ) 1

X

X

R

i f x for all x R and ii f x dx

( ) ( ) ( )b

a

P a X b P a X b f x dx


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Cumulative Distribution Function (cdf)

If X is an RV, discrete or continuous, then P(Xx) is calledthe cumulative distribution function of X or distributionfunction of X and denoted as F(x).

( )

( )

j

j

X x

x

p if X is discrete

F x

f x dx if X is continuous


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Properties of cdf

F(x) is a non decreasing function of x. i.e if x1 < x2, thenF(x1)F(x2).

F(-) = 0 and F() = 1

If X is a discrete RV taking values of x1, x2, where

x1


6/12

Expectation of RV, Mean and Variance

( )

( )

j

j

xxp if X is discrete

E X

xf x dx if X is continuous

Properties of Expectation

1. E(constant) = constant

2. E(aX) = a E(X)3. E(aX bY ) = a E(X) b E(Y) if X and Y are

independent.


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Properties of Variance

Variance of X = Var(X) = V(X) = E(X2)[E(X)]2

1. V(constant) = 02. V(aX) = a2V(X)

3. V(aX bY) = a2V(X) + b2V(Y)

4. Variance cannot be negative

Moments

The rth raw moment is denoted as and rth centralmoment is denoted as r.

E(X )rr

r


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( )

( )( ) ( )

j

r

j

xth r

r

x A p if X is discrete

r moment about a value A E X Ax A f x dx if X is continuous

( 0)

( 0)

( 0) ( )

j

r

j

xth r

rr

x p if X is discrete

r moment about origin E X

x f x dx if X is continuous

( )

( )

( ) ( )

j

r

j

xth r

r

x x p if X is discrete

r moment about mean x E X x

x x f x dx if X is continuous


9/12

Moment Generation Function (MGF)

( ) ( )

( )

j

tXj

X xtX

xtX

e p if X is discrete

M t E e

e f x dx if X is continuous

2 3

1 2 3( ) ( ) 1 ... ...2! 3! !

rt X

X rt t tM t E e t

r

2

1 2 2

0 0 0

( ) ( ) ( )r

X X r Xr

t t t

d d dM t M t M tdt dt dt


10/12

Relationship between raw moments and centralmoments

1 1

2

2 2 1

3

3 3 2 1 12 4

4 4 3 1 2 1 1

( )

3 2 ( )

4 6 ( ) 3( )

A


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Problems

1. A shipment of 6 television sets contains 2 defective sets. A hotel makes a random purchaseof 3 of the sets. If X is the number of defective sets purchased by the hotel, find theprobability distribution of X. Ans: P(0) = P(2) = 1/5 and P(1) = 3/5

2. A random variable X may assume 4 values with probabilities (1+3x)/4, (1-x)/4, (1+2x)/4and (1-4x)/4. Find the condition on x so that these values represent the probabilityfunction of X? Ans: -1/3 x 1/4

3. If the RV X takes the values 1,2,3 and 4 such that 2P(X=1) =3P(X=2) =P(X=3)=5P(X=4), find the probability distribution function and cumulative distribution functionof X.

4. A RV X has the following probability distributionx -2 -1 0 1 2 3p(x) 0.1 K 0.2 2K 0.3 3K

(a) Find K (b) evaluate P(X < 2) and P(-2 < X < 2) (c) find the cdf of X.5. The probability function of an infinite discrete distribution is given by

P(X = j) =1/kj(j = 1,2,). Find the value of k, and find the mean and variance of the

distribution. Find also P( X is even), P(X 5) and P( X is divisible by 3).

Ans: 2, 2,2, 1/3, 1/16, 1/76. A RV X has the following probability distribution

x 0 1 2 3 4 5 6 7p(x) 0 K 2K 2K 3K K2 2K2 7K2 + KFind (i) the value of K, (ii) P(1.5 < X < 4.5 / X > 2) and (iii) the smallest value of suchthat P(X ) > .

2


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1. If (a) show that f(x) is a pdf (b) find its

distribution function F(x).

2. If the density function of a RV is given by

(a) Find the value of a(b) Find the cdf of X

3. A continuous RV X that can take assume any value between x = 2 and x = 5 has adensity function given by f(x) = k (1 + x). Find P(X < 4).

4. A RV X has a pdf f(x) = k x2 e-x, x > 0. Find k, mean and variance.

5. The distribution function of a RV is X is given by F(x) = 1(1 + x) e-x, x 0. Findthe pdf, mean and variance of X.

6. The cdf of a continuous RV X is given by Find the pdf of X and evaluate

and

using both the pdf and cdf.

2

2 0( )0 0

x

xe xf xx

, 0 1

, 1 2( )

3 , 2 3

0,

ax x

a xf x

a ax x

elsewhere

2

2

0, 01

, 02

( )3 1

1 (3 ), 325 2

1, 3

x

x x

F x

x x

x

( 1)P X

14

3P X

unit i - random variables

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