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Discrete Random Variables
Distribution, Expectation, Variance
Random Variable
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A random variable is a function whose domain is the sample space and
whose range is a subset of the set of real numbers.
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A random variable is a function that associates a unique numerical value with every outcome of an experiment. The value of the random variable will vary from trial to trial as the experiment is repeated. * Experiment : Toss a coin twice Sample Space : {HH, HT, TH, TT} X = number of heads X(HH) = 2, X(HT) = 1, X(TH) = 1, X(TT) = 0 X :
Ω
{0,1,2} is a function. Hence X is a random variable (*http://www.stats.gla.ac.uk/steps/glossary/probability_distributions.html#randvar )
Experiment : Draw three balls at random from an urn containing two red and two green balls.
Sample Space = {RRG, RGG} X = number of red balls X(RRG) = 2, X(RGG) = 1 X :
Ω
{1,2} is a function. Hence X is a random variable. Experiment : Toss two dice.
Sample Space = {(1,1), (1,2),…, (1,6),…, (6,1),…(6,6)}
X = sum of the scores obtained in the two tosses
X((1,1)) = 2, X((1,2)) = 3, …, X((6,6)) = 12
X :
Ω
{2, …,12} is a function.
X is a random variable
Discrete Random Variable
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A random variable is said to be discrete if the range of values taken by the random variable is either finite or is countably infinite.
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All the examples of random variables given till now are discrete random variables
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An example of a random variable which is not discrete is given below : Experiment : Choose three students at random from PGP-1 Sample Space : {{a, b, c}: a, b, c are students of PGP-1} X = total sum of weights of the three students X can take any positive value. Hence X is not a discrete random variable.

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