Probability Terminology and Concepts

8/8/2019 Probability Terminology and Concepts

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Discrete Random VariablesDefn: Let X be a Random variable. If the no:

of possible values of X is finite or countably

infinite,X is called a discrete random

variable. The possible values of X may be

listed as x1,x2,..In the finite case the

list terminates . In the countably infinitecase the list continues indefinitely.


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Example 1:

The no: of jobs arriving each week at a job shop is

observed.The random variable of interest is X,

whereX=Num of jobs arriving each week

The possible values of X are given by the rangespace of X,which is denoted by Rx.

Here Rx={0,1,2,..}


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With each possible xi in Rx, a number p(xi)=P(X=xi)gives the probability that the random variableequals the value x

i.The numbers p(xi),i= 1,2,,must satisfy thefollowing conditions.

1. p(xi)>=0 for all i

2. i=1p(xi)=1

The collection of pairs (xi,p(xi)) ,i=1,2is called theprobability distribution of X.


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p(xi) is called theprobability mass function

(pmf) of X.


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Example 2

Experiment of tossing a single die.

X: Num of spots on the up face of the die after the

toss

Rx={1,2,3,4,5,6}

Assume the die is loaded so that the probability that

a given face lands up is proportional to the num ofspots showing.

Then,


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The discrete probability distribution for this random

experiment is given by

xi 1 2 3 4 5 6

p(xi)1/21 2/21 3/21 4/21 5/21 6/21


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The earlier stated conditions are satisfied .

ie, p(xi)>=0 for i= 1,2,.6

i

=1p(xi) = 1/21+.+6/21 = 1


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Continuous Random Variable

If the range space Rx of the random variable is an

interval or a collection of intervals,X is called a

continuous random variable.For a continuous random variable X,the probability

that X lies in the interval [a,b] is given by

P(a


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The pdf satisfies the following conditions:

(a) f(x)>=0 for all x in Rx

(b) Rxf(x)dx=1

(c) f(x)=0 if x is not in Rx

For any specified value x0,P(X=x0)=0 since

f(x)dx=0


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Example

The life of a Laser ray device used to inspect thecracks in aircraft wings is given by X,a continuous

random variable assuming all values in the rangex>=0.

The pdf of the life time , in years is as follows

f(x) = e-x/2 ,x>=0

0 , otherwiseThe random variable X is said to have an exponential

distribution with mean 2.


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The probability that the laser ray device is between 2

and 3 years is determined from

P(2


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Shashwat [email protected]

Probability Terminology and Concepts

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