Basic Concepts in Probability Basic Probability Rules Connecting Probability to Sampling.
Probability Terminology and Concepts
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Transcript of 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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