An automatic lathe machine produces on an average 2 perfect pins a day. a. What

Question

An automatic lathe machine produces on an average 2 perfect pins a day.

a. What is the probability that there will be at least 3 perfect pins on a given day?

b. What is the probability that there will be no perfect pins produced on a given day?

c. What is the probability that there will be at most 4 perfect pins produced on a given day?

d. Consider the number of perfect pins produced on three consecutive days. Let ‘T” be the total number of perfect pins produced in these 3 days. What is the expectation of T? What is the standard deviation of T?

Explanation / Answer

Possion Distribution
PMF of P.D is = f ( k ) = e- x / x!
Where   
= parameter of the distribution.
x = is the number of independent trials
a.
P( X < 3) = P(X=2) + P(X=1) + P(X=0)
= e^-2 * 0 ^ 2 / 2! + e^-2 * ^ 1 / 1! + e^-2 * ^ 0 / 0!
= 0.6767
P( X > = 3 ) = 1 - P (X < 3) = 0.3233
b.
P( X = 0 ) = e ^-2 * 2^0 / 0! = 0.1353
c.
P( X < = 4) = P(X=4) + P(X=3) + P(X=2) + P(X=1) + P(X=0)   
= e^-2 * 2 ^ 4 / 4! + e^-2 * 0 ^ 3 / 3! + e^-2 * ^ 2 / 2! + e^-2 * ^ 1 / 1! + e^-2 * ^ 0 / 0!
= 0.9473

d.
Total of no.of pins produce in 2 days = 3 * 2 = 6 perfect pins

expectation of T = 6

standard deviation of T = Sqrt(6) = 2.4494

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