Answer to Question #330730 in Statistics and Probability for almen

Question #330730

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
2022-04-19T05:07:31-0400

It has to be pointed out that the distribution is not specified. Suppose that XX is a random variable that corresponds to the 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 with parameters μ=4\mu=4 and σ\sigma . Then, P(X>2)=2+f(x)dxP(X>2)=\int_{2}^{+\infty}f(x)dx, where f(x)=1σ2πe12(xμσ)2f(x)=\frac{1}{\sigma\sqrt{2\pi}}e^{-\frac12\left(\frac{x-\mu}{\sigma}\right)^2}. For σ=1\sigma=1 we get: P(X>2)0.977.P(X>2)\approx0.977.
  2. Suppose that XX has a Poisson distribution with λ=4.\lambda=4. Remind that for Poisson distribution P(X=k)=λkeλk!P(X=k)=\frac{\lambda^ke^{-\lambda}}{k!}. Thus, P(X>2)=P(X=3)+P(X=4)+...0.762P(X>2)=P(X=3)+P(X=4)+...\approx0.762

Thus, for the normal distribution we receive 0.9770.977 and for the Poisson distribution 0.7620.762.


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