Answer to Question #87028 in Statistics and Probability for Swati malik

Question #87028
The number of cars arriving at a railway station follows the poisson distribution. If the average number of cars arriving during a specified period of half an hour is 2. Find the probabilities that during a give half an hour.
i. No car will arrive
ii. At least two cars will arrive
iii. At the most 3 cars will arrive
iv. Between 1 and 3 cars will arrive
1
Expert's answer
2019-03-26T12:15:22-0400

The probability distribution of the Poisson random variable X, representing the number of outcomes occurring

in a given time interval or specified region denoted by t, is


"p(x; \\lambda t)={e^{-\\lambda t}(\\lambda t)^x \\over {x!}}"

Use the Poisson distribution with 


"\\lambda t=2"

i. No car will arrive 


"P(X=0)={e^{-2}(2)^0 \\over {0!}}=e^{-2}\\approx0.135335"

ii. At least two cars will arrive 


"P(X\\ge2)=1-(P(X=0)+P(X=1))="

"=1-({e^{-2}(2)^0 \\over {0!}}+{e^{-2}(2)^1 \\over {1!}})=1-e^{-2}-2e^{-2}=1-3e^{-2}\\approx0.593994"

iii. At the most 3 cars will arrive 


"P(X\\le3)=P(X=0)+P(X=1)+P(X=2)+P(X=3)="

"={e^{-2}(2)^0 \\over {0!}}+{e^{-2}(2)^1 \\over {1!}}+{e^{-2}(2)^2 \\over {2!}}+{e^{-2}(2)^3 \\over {3!}}="

"={19 \\over 3}e^{-2}\\approx0.857123"

iv. Between 1 and 3 cars will arrive


"P(1\\le X\\le3)=P(X=1)+P(X=2)+P(X=3)="

"={e^{-2}(2)^1 \\over {1!}}+{e^{-2}(2)^2 \\over {2!}}+{e^{-2}(2)^3 \\over {3!}}={16 \\over 3}e^{-2}\\approx0.721788"




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