A manuscript is sent to a typing firm consisting of typists A, B and C. If it is typed by A, then the number of errors made is a Poisson random variable with mean 2.6; if typed by B then the number of errors made is a Poisson random variable with mean 3; and if typed by C then it is a Poisson random variable with mean 3.4. Let X denote the number of errors in the manuscript. Assume that each typist is equally likely to do the work. (a) Find E (X) (b) Find Var (X)

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7 June, 22:00

A manuscript is sent to a typing firm consisting of typists A, B and C. If it is typed by A, then the number of errors made is a Poisson random variable with mean 2.6; if typed by B then the number of errors made is a Poisson random variable with mean 3; and if typed by C then it is a Poisson random variable with mean 3.4. Let X denote the number of errors in the manuscript. Assume that each typist is equally likely to do the work. (a) Find E (X) (b) Find Var (X)

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Zayne Reynolds · Answer 1

(a) E (X) = 3

(b) Var (X) = 12.1067

Explanation:

(a) E[X]

E[X]T = E[X]T=A + E[X]T=B + E[X]T=C

= (2.6 + 3 + 3.4) / 3

= 2.6 (1/3) + 3 (1/3) + 3.4 (1/3)

= 2.6/3 + 1 + 3.4/3

= 3

(b) Var (X) = E[X²] - (E[X]) ²

Recall that if Y ∼ Pois (λ), then E[Y 2] = λ+λ2. This implies that

E[X²] = [ (2.6 + 2.6²) + (3 + 3²) + (3.4 + 3.4²) ]/3

= (9.36 + 12 + 14.96) / 3

= 36.32/3

= 12.1067

Var (X) = E[X²] - (E[X]) ²

= 12 - 3²

= 12.1067 - 9

= 3.1067