c) D 73 a) Define a discrete r.v. Giving illustrations, explain what is meant by
ID: 3245945 • Letter: C
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c) D 73 a) Define a discrete r.v. Giving illustrations, explain what is meant by a discrete p.d. b) A bag contains 4 red and 6 black balls. A sample of 4 balls is selected from the bag without replacement, Let X be the number of red balls. Find the p.d. for X. (P.U., B.A/B.Sc. 1975) A large store places its last 15 clock radios in a clearance sale. Unknown to any one, 5 of radios are defective. If a customer tests 3 different clock radios selected at random, whatis the p.d. of X- number of defective radios in the sample? Three balls are drawn from a bag containing 5 white and 3 black balls. If X denotes the number of white balls drawn from the bag, then find the p.d. of X earance sale. Unknown to any one, 5 of the erent clock radios selected at random, what i 7.4 a) b)Explanation / Answer
Part 7.3(a)
A discrete random variable is a variable which can take values only at discrete intervals and cannot take any value in between.
Examples:
Number of accidents that occur in a particular stretch of road during 9 to 10 AM. Here, the variable can take only values 0, 1, 2, 3, ………, mathematically it can go up to infinity, but it cannot be 0.5 or 1.2 or 3.7 etc.
Number of defectives in a sample of, say 5. Here the values can only be 0, 1, 2, 3, 4 or 5 and not ½ or 5/4 or 19/4 etc.
[Note: Additional input: Guard against most popular error: it is not necessary that the values taken by a discrete variable are necessarily integers. For example, imagine a man giving his daughter pocket money everyday in multiples of half a dollar. Here, the values taken can be ½, 1, 3/2, 2, 5/2, …. etc. However, if we change the language and say the man gives in multiples of 50 cents, then the values are integers: 50, 100, 150, …. etc.]
p.d. or probability density, of a discrete random variable is the mathematical function which gives a formula for finding the probability for the various values taken by the variable. i.e., if X is a discrete random variable taking values, say x1, x2, ….., xn, then pd of X is given by
p(xi) = P(X = xi), i = 1, 2, …., n.
Part 7.3(a)
Let X = number of red balls in a without-replacement-sample of 4 taken from a bag containing 4 red and 6 black balls.
Then, X can take values 0. 1, 2, 3 or 4.
P(X = xi) = (4Cxi)(6C4-xi)/(10C4) or in general,
pd(X) = p(x) = (4Cx)(6C4-x)/(10C4), x = 0. 1, 2, 3, 4 ANSWER
[Explanation for (4Cxi)(6C4-xi)/(10C4): if xi are red, then (4 – xi) must be black. xi red can be selected from 4 red in (4Cxi) and (4 – xi) black can be selected from 6 black in (6C4-xi). A sample of 4 from 10 can selected in (10C4) ways ].
[Additional input: This distribution is known as Hyper-geometric Distribution]
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