Answer to Question #334828 in Statistics and Probability for bea villamar

Question #334828

the average number of automobiles per minute stopping for gas at a particular service station along the coastal road is 4 what is the probability that in any given minute more than two will stop for gas


1
Expert's answer
2022-04-29T09:07:35-0400

It has to be pointed out that the distribution is not specified. Suppose that XX is a random variable that corresponds to the average number of automobiles per minute. The aim is to compute P(X>2)P(X>2). Consider the most typical distributions:

1. Suppose that XX has a normal distribution :XN(μ,σ2).X\sim N(\mu, \sigma^2).

Then


P(X>2)=1P(X2)=1P(Z241)P(X>2)=1-P(X\le2)=1-P(Z\le\dfrac{2-4}{1})

=1P(Z2)0.97725=1-P(Z\le-2)\approx0.97725


2. Suppose that XX has a Poisson distribution with λ=4:XPo(λ).\lambda=4:X\sim Po(\lambda).

Then


P(X>2)=1P(X=0)P(X=1)P(X>2)=1-P(X=0)-P(X=1)

P(X=2)=1e4(4)00!e4(4)11!-P(X=2)=1-\dfrac{e^{-4}(4)^0}{0!}-\dfrac{e^{-4}(4)^1}{1!}

e4(4)22!0.76190-\dfrac{e^{-4}(4)^2}{2!}\approx0.76190


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