An internet search engine looks for a certain keyword in a sequence of independe

Question

An internet search engine looks for a certain keyword in a sequence of independent web sites. It is believed that 20% of the sites contain this keyword.

(a) Let X be the number of websites visited until the first keyword is found. Find the distribution of X. (

b) Compute the expected value and the standard deviation of X.

(c) Out of the first 10 websites, let Y be the number of sites that contain the keyword. Find the distribution of Y .

(d) Compute the expected value and the standard deviation of Y .

(e) Compute the probability that at least 3 of the first 10 websites contain the keyword.

Explanation / Answer

Solution:

Given P =20% = 0.2
a) P(X = x) = q^x-1 p^1 = (0.8)^x-1 (0.2)^1 (here x = 1,2,3,...)
b) q/p = 0.8/0.2 = 4
Standard deviation = sqrt(q/p^2)
= sqrt(0.8/0.2^2)
= 4.4721
c) P(Y = y) = nCy p^y q^n-y
= 10Cy (0.2)^y (0.8)^10-y
d) Expected value = np = 10(0.2) = 2
Standard deviation = sqrt(npq) = sqrt(10(0.2)(0.8))
= 1.2649
e) P(at least 3) = P(Y 3)
= 1-P(Y < 3)
= 1-{P(Y = 0) + P(Y = 1) + P( Y = 2) }
= 1-{ 10C0 (0.2)^0 (0.8)^10-0 + 10C1 (0.2)^1 (0.8)^10-1+ 10C2 (0.2)^2 (0.8)^10-2}
= 1-{0.1073742 + 0.2684355 + 0.3019899}
= 1- 0.6777996
= 0.3222004

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