Chapter: Mathematics (maths) - Random Variables

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Random Variables

1 Introduction 2 Discrete Random Variables 3 Continuous Random Variables 4 Moments 5 Moment generating functions 6 Binomial distribution 7 Poisson distribution 8 Geometric distribution 9 Uniform distribution 10 Exponential distribution 11 Gamma distribution

UNIT - I

Random Variables

 

1        Introduction

2        Discrete Random Variables

3        Continuous Random Variables

4        Moments

5        Moment generating functions

6        Binomial distribution

7        Poisson distribution

8        Geometric distribution

9        Uniform distribution

10      Exponential distribution

11      Gamma distribution

 

 

Introduction

 

Consider an experiment of throwing a coin twice. The outcomes {HH, HT, TH, TT} consider the sample space. Each of these outcome can be associated with a number by specifying a rule of association with a number by specifying a rule of association (eg. The number of heads). Such a rule of association is called a random variable. We denote a random variable by the capital letter (X, Y, etc) and any particular value of the random variable by x and y.

 

Thus a random variable X can be considered as a function that maps all elements in the sample space S into points on the real line. The notation X(S)=x means that x is the value associated with the outcomes S by the Random variable X.

 

1 SAMPLE SPACE

 

Consider an experiment of throwing a coin twice. The outcomes S = {HH, HT, TH, TT} constitute the sample space.

 

2 RANDOM VARIABLE

 

In this sample space each of these outcomes can be associated with a number by specifying a rule of association. Such a rule of association is called a random variables.

Eg : Number of heads

 

We denote random variable by the letter (X, Y, etc) and any particular value of the random variable by x or y.

S = {HH, HT, TH, TT} X(S) = {2, 1, 1, 0}

Thus a random X can be the considered as a fun. That maps all elements in the sample space S into points on the real line. The notation X(S) = x means that x is the value associated with outcome s by the R.V.X.

Example

In the experiment of throwing a coin twice the sample space S is S = {HH,HT,TH,TT}. Let X be a random variable chosen such that X(S) = x (the number of heads).

 

Note

Any random variable whose only possible values are 0 and 1 is called a Bernoulli random variable.

 

2.1 DISCRETE RANDOM VARIABLE

Definition : A discrete random variable is a R.V.X whose possible values consitute finite set of values or countably infinite set of values.

Examples

All the R.V.’s from Example : 1 are discrete R.V’s

Remark

The meaning of P(X ≤a).

 

P(X ≤a) is simply the probability of the set of outcomes ‘S’ in the sample space for which X(s) ≤ a.

Or

P(X≤a) = P{S : X(S) ≤ a}

In the above example : 1 we should write

P(X ≤ 1) = P(HH, HT, TH) = ¾

Here P(X≤1) = ¾ means the probability of the R.V.X (the number of heads) is less than or equal to 1 is ¾ .

 

 

Distribution function of the random variable X or cumulative distribution of the random variable X

Def :

 

The distribution function of a random variable X defined in (-∞, ∞) is given by F(x) = P(X ≤ x) = P{s : X(s) ≤ x}

Note

 

Let the random variable X takes values x1, x2, ….., xn with probabilities P1, P2, ….., Pn and let x1< x2< ….. <xn

 

Then we have                

F(x)   = P(X < x1) = 0, -∞ < x < x,            

F(x)   = P(X < x1) = 0, P(X < x1) + P(X = x1)     = 0 + p1 = p1

F(x)   = P(X < x2) = 0, P(X < x1) + P(X = x1) + P(X = x2) = p1 + p2

F(x)   = P(X < xn) = P(X < x1) + P(X = x1) + ….. + P(X = xn)

          = p1 + p2+ ………. + pn        = 1   

 

2.2 PROPERTIES OF DISTRIBUTION FUNCTIONS

Property : 1          P(a < X ≤ b)  = F(b) – F(a), where F(x) = P(X ≤ x)

Property : 2          P(a ≤ X ≤ b)  = P(X = a) + F(b) – F(a)

Property : 3          P(a < X < b)  = P(a < X ≤ b) - P(X = b)

= F(b) – F(a) – P(X = b) by prob (1)

 

2.3 PROBABILITY MASS FUNCTION (OR) PROBABILITY FUNCTION

 

Let X be a one dimenstional discrete R.V.  which          takes the values x1, x2, …… To each possible outcome ‘xi’ we can associate a number pi.

i.e.,    P(X = xi) = P(xi)  =  pi   called  the   probability  of xi. The number pi = P(xi) satisfies the following conditions.


 

The function p(x) satisfying the above two conditions is called the probability mass function (or) probability distribution of the R.V.X. The probability distribution {xi, pi} can be displayed in the form of table as shown below.

 


 

Notation

 Let ‘S’ be a sample space. The set of all outcomes ‘S’ in S such that X(S) = x is denoted by writing X = x.

 P(X = x) = P{S : X(s) = x}

|||ly P(x ≤ a) = P{S : X() (-∞, a)}

and P(a < x ≤ b) = P{s : X(s) (a, b)}

P(X = a or X = b) = P{(X = a) (X = b)}

P(X = a and X = b) = P{(X = a) ∩ (X = b)} and so on.

 

Theorem :1 If X1 and X2 are random variable and K is a constant then KX1, X1 + X2, X1X2, K1X1 + K2X2, X1-X2 are also random variables.

 

Theorem :2

 

If ‘X’ is a random variable and f(•) is a continuous function, then f(X) is a random variable.

 

Note

If F(x) is the distribution function of one dimensional random variable then


 

Example:

A random variable X has the following probability function


(i)      Determine the value of ‘a’

(ii)     Find P(X<3), P(X≥3), P(0<X<5)

(iii)    Find the distribution function of X.

 

Solution

Table 1


(i)                We know that if p(x) is the probability of mass function then


 

 

 

3 CONTINUOUS RANDOM VARIABLE

Def : A R.V.’X’ which takes all possible values in a given internal is called a continuous random variable.

Example : Age, height, weight are continuous R.V.’s.

 

3.1 PROBABILITY DENSITY FUNCTION

 

Consider a continuous R.V. ‘X’ specified on a certain interval (a, b) (which can also be a infinite interval (-∞, ∞)).

If there is a function y = f(x) such that


 

Then this function f(x) is termed as the probability density function (or) simply density function of the R.V. ‘X’.

It is also called the frequency function, distribution density or the probability density function.

The curve y = f(x) is called the probability curve of the distribution curve.

Remark

 

If f(x) is p.d.f of the R.V.X then the probability that a value of the R.V. X will fall in some interval (a, b) is equal to the definite integral of the function f(x) a to b.


 

3.2 PROPERTIES OF P.D.F

 

The p.d.f f(x) of a R.V.X has the following properties


1.       In the case of discrete R.V. the probability at a point say at x = c is not zero. But in the case of a continuous R.V.X the probability at a point is always zero.


 

2.     If x is a continuous R.V. then we have p(a ≤ X ≤ b) = p(a ≤ X < b)

 

= p(a < X V b)

 

IMPORTANT DEFINITIONS INTERMS OF P.D.F

 

If f(x) is the p.d.f of a random variable ‘X’ which is defined in the interval (a, b) then


 

3.3 Mathematical Expectations

 

Def :Let ‘X’ be a continuous random variable with probability density function f(x). Then the mathematical expectation of ‘X’ is denoted by E(X) and is given by

 




3.4 EXPECTATIONS (Discrete R.V.’s)

Let ‘X’ be a discrete random variable with P.M.F p(x)

Then


 

3.5 ADDITION THEOREM (EXPECTATION)

Theorem 1

 

If X and Y are two continuous random variable with pdf fx(x) and fy(y) then

 

E(X+Y) = E(X) + E(Y)

 

3.6 MULTIPLICATION THEOREM OF EXPECTATION

Theorem 2

 

If X and Y are independent random variables,

Then E(XY) = E(X) . E(Y)

Note :

If X1, X2, ……, Xn are ‘n’ independent random variables, then

 

E[X1, X2, ……, Xn] = E(X1), E(X2), ……, E(Xn)

 

Theorem 3

If ‘X’ is a random variable with pdf f(x) and ‘a’ is a constant, then

 

(i)                E[a G(x)] = a E[G(x)]

 

(ii)             E[G(x)+a] = E[G(x)+a]

 

Where G(X) is a function of ‘X’ which is also a random variable.

 

Theorem 4

 

If ‘X’ is a random variable with p.d.f. f(x) and ‘a’ and ‘b’ are constants, then E[ax + b] = a E(X) + b

 

Cor 1:

 

If we take a = 1 and b = –E(X) = – X , then we get

E(X- X ) = E(X) – E(X) = 0

Note


 

3.7 EXPECTATION OF A LINEAR COMBINATION OF RANDOM VARIABLES

 

Let X1, X2, ……, Xn be any ‘n’ random variable and if a1, a2 , ……, an are constants, then E[a1X1 + a2X2 + ……+ anXn] = a1E(X1) + a2E(X2)+ ……+ anE(Xn)

Result

If X is a random variable, then

Var (aX + b) = a2Var(X) ‘a’ and ‘b’ are constants.

 

Covariance :

 

If X and Y are random variables, then covariance between them is defined as Cov(X, Y) = E{[X - E(X)] [Y - E(Y)]}

          Cov(X, Y)   =       E(XY) – E(X) . E(Y)      (A)

If X and Y are independent, then               

E(XY)         = E(X) E(Y)                            

Sub (B) in (A), we get Cov (X, Y) = 0

 

If X and Y are independent, then

Cov (X, Y) = 0                                                    

Note                                                           

          (i)      Cov(aX, bY)        = ab Cov(X, Y)                      

          (ii)     Cov(X+a, Y+b)    = Cov(X, Y)                           

          (iii)    Cov(aX+b, cY+d) = ac Cov(X, Y)             

          (iv)    Var (X1 + X2) = Var(X1) + Var(X2) + 2 Cov(X1, X2)

If X1, X2 are independent                                            

          Var (X1+ X2) = Var(X1) + Var(X2)                            

 

EXPECTATION TABLE

 


 

SOLVED PROBLEMS ON DISCRETE R.V’S

Example :1

When die is thrown, ‘X’ denotes the number that turns up. Find E(X), E(X2) and Var (X).

Solution

 

Let ‘X’ be the R.V. denoting the number that turns up in a die. ‘X’ takes values 1, 2, 3, 4, 5, 6 and with probability 1/6 for each

 

 

 



4 CONTINUOUS DISTRIBUTION FUNCTION

Def :

 

If f(x) is a p.d.f. of a continuous random variable ‘X’, then the function


 

is called the distribution function or cumulative distribution function of the random variable.

* PROPERTIES OF CDF OF A R.V. ‘X’




 

 



 

5 MOMENT GENERATING FUNCTION

 

Def : The moment generating function (MGF) of a random variable ‘X’ (about origin) whose probability function f(x) is given by


 

 






6 Discrete Distributions

 

The important discrete distribution of a random variable ‘X’ are

1.     Binomial Distribution

2.     Poisson Distribution

 

3.     Geometric Distribution

6.1BINOMIAL DISTRIBUTION

 

Def : A random variable X is said to follow binomial distribution if its probability law is given by

 

P(x) = p(X = x successes) = nCx px qn-x Where x = 0, 1, 2, ……., n, p+q = 1

Note

Assumptions in Binomial distribution

i)         There are only two possible outcomes for each trail (success or failure).

ii)       The probability of a success is the same for each trail.

 

iii)    There are ‘n’ trails, where ‘n’ is a constant.

 

iv)    The ‘n’ trails are independent.

 

 

Example :1.6.1

Find the Moment Generating Function (MGF) of a binomial distribution about origin.

 

Solution



 

 

 

7 Passion Distribution

Def :

 

A random variable X is said to follow if its probability law is given by


 

Poisson distribution is a limiting case of binomial distribution under the following conditions or assumptions.

1.       The number of trails ‘n’ should e infinitely large i.e. n→∞.

2.       The probability of successes ‘p’ for each trail is infinitely small.

3.       np = λ , should be finite where λ is a constant.

 

* To find MGF



 PROBLEMS ON POISSON DISTRIBUTION





7.3 Derive probability mass function of Poisson distribution as a limiting case of Binomial distribution

Solution

 

We know that the Binomial distribution is P(X=x) = nCx pxqn-x


 

 

 


8 GEOMETRIC DISTRIBUTION

Def: A discrete random variable ‘X’ is said to follow geometric distribution, if it assumes only  non-negative values and its probability mass function is given by








9 CONTINUOUS DISTRIBUTIONS

 

If ‘X’ is a continuous random variable then we have the following distribution

1.     Uniform (Rectangular Distribution)

2.     Exponential Distribution

3.     Gamma Distribution

 4.     Normal Distribution

 

 

9.1 Uniform Distribution (Rectangular Distribution)

 

Def : A random variable X is set to follow uniform distribution if its



 



PROBLEMS ON UNIFORM DISTRIBUTION

Example 1.9.1

If X is uniformly distributed over (-α,α), α< 0, find α so that

(i)      P(X>1) = 1/3

(ii)     P(|X| < 1) = P(|X| > 1)

Solution

If X is uniformly distributed in (-α, α), then its p.d.f. is


 

 

10 THE EXPONENTIAL DISTRIBUTION

 

Def :A continuous random variable ‘X’ is said to follow an exponential distribution with parameter λ>0 if its probability density function is given by





 

 

 

11 GAMMA DISTRIBUTION

Definition

 

A Continuous random variable X taking non-negative values is said to follow gamma distribution , if its probability density function is given by



When α is the parameter of the distribution.

Additive property of Gamma Variates

If X1,X2 , X3,.... Xk are independent gamma variates with parameters λ1,λ2,…..λ krespectively then X1+X2 + X3+.... +Xk is also a gamma variates with parameter λ1+ λ2 +….. + λk

 

Example :1.11.1

Customer demand for milk in a certain locality ,per month , is Known to be a general Gamma RV.If the average demand is a liters and the most likely demand b liters (b<a) , what is the varience of the demand?

 

Solution :

 

Let X be represent the monthly Customer demand for milk. Average demand is the value of E(X).

Most likely demand is the value of the mode of X or the value of X for which its density function is maximum.

If f(x) is the its density function of X ,then


 

 

 

 

TUTORIAL QUESTIONS

 

1.It is known that the probability of an item produced by a certain machine will be defective is 0.05. If the produced items are sent to the market in packets of 20, fine the no. of packets containing at least, exactly and atmost 2 defective items in a consignment of 1000 packets using (i) Binomial distribution (ii) Poisson approximation to binomial distribution.

2. The daily consumption of milk in excess of 20,000 gallons is approximately exponentially distributed with . 3000 = θ The city has a daily stock of 35,000 gallons. What is the probability that of two days selected at random, the stock is insufficient for both days.

3.The density function of a random variable X is given by f(x)= KX(2-X), 0≤X≤2.Find K, mean, variance and rth moment.

4.A binomial variable X satisfies the relation 9P(X=4)=P(X=2) when n=6. Find the parameter p of the Binomial distribution.

5.       Find the M.G.F for Poisson Distribution.

6.       If X and Y are independent Poisson variates such that P(X=1)=P(X=2) and P(Y=2)=P(Y=3). Find V(X-2Y).

7.A discrete random variable has the following probability distribution   

X:      0        1        2        3        4        5        6        7        8

P(X)  a        3a      5a      7a      9a      11a    13a    15a    17a

Find the value of a, P(X<3) and c.d.f of X.                            

7. In a component manufacturing industry, there is a small probability of 1/500 for any component to be defective. The components are supplied in packets of 10. Use Poisson distribution to calculate the approximate number of packets containing (1). No defective. (2). Two defective components in a consignment of 10,000 packets. 




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