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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
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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
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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
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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