Question

The number of traffic lights malfunctioning daily in any city can be said to satisfy the binomial distribution. However, the rate at which “successes” (traffic lights malfunctioning) and “failures” (traffic lights not malfunctioning) occurs nearly instantaneously, with nearly nonzero probability, such that the expected value approaches a constant. It is known that the second moments (E[X2]) of the number of daily malfunctioning traffic lights in Philadelphia, Pittsburgh, and Erie respectively are 72, 56, and 42. The malfunctioning of traffic lights between different cities is mutually independent.

Let ? = # of malfunctioning traffic lights during a day in Philadelphia, ? = # of malfunctioning traffic lights during 2 days in Pittsburgh, and ? = # of malfunctioning traffic lights during a day in Erie. Calculate P(? + ? + ? = 24).

Answer 1

Solution

Back-up Theory

If an event has infinite number of possibilities of occurrence, but the actual probability of occurrence is very low, such that the expected value approaches a constant, then the number of times the event occurs follows Poisson ............ (1)

If a random variable X ~ Poisson (λ), i.e., X has Poisson Distribution with mean λ then

probability mass function (pmf) of X is given by P(X = x) = e ^{– λ}.λ^x/(x!) ……................................................................……..(2)

This probability can also be obtained by using Excel Function, Statistical, POISSON …................................................... (2a)

Mean = λ...............................................................................................................................................................................(3a)

Variance = λ..........................................................................................................................................................................(3b)

Standard deviation = √λ .......................................................................................................................................................(3c)

Variance = E(X²) – {E(X)}² i.e., E(X²) = λ + λ2 = λ(λ + 1)...................................................................................................(3d)

If X = number of times an event occurs during period t, Y = number of times the same event

occurs during period kt, and X ~ Poisson(λ), then Y ~ Poisson (kλ) ………..............................................................…….. (4)

If X₁ ~ Poisson (λ₁), X₂ ~ Poisson (λ₂) and X₁, X₂ are independent, then (X₁ + X₂) ~ Poisson (λ₁ + λ₂)........................... (5)

If X₁ ~ Poisson (λ₁), X₂ ~ Poisson (λ₂), …….., X_k ~ Poisson (λ_k) and X₁, X₂, …….., X_k are independent,

then (X₁ + X₂ + …….., + X_k) ~ Poisson (λ₁ + λ₂ + …… + λ_k)………....................................…................…………………..(5a)

Now to work out the solution,

Given, ‘the rate at which “successes” (traffic lights malfunctioning) and “failures” (traffic lights not malfunctioning) occurs nearly instantaneously, with nearly nonzero probability, such that the expected value approaches a constant.’ Vide (1), # of malfunctioning traffic lights during a day is Poisson..................................................................................... (6)

Also given ‘that the second moments (E[X2]) of the number of daily malfunctioning traffic lights in Philadelphia, Pittsburgh, and Erie respectively are 72, 56, and 42.’ vide (3d),

λ for Philadelphia, say λ1 = 8, ......................................................................................................................................... (7a)

λ for Pittsburgh, say λ2 = 7, ............................................................................................................................................ (7b)

λ for Erie, say λ3 = 6, ...................................................................................................................................................... (7c)

Further given that, ‘? = # of malfunctioning traffic lights during a day in Philadelphia, ? = # of malfunctioning traffic lights during 2 days in Pittsburgh, and ? = # of malfunctioning traffic lights during a day in Erie.’

X ~ Poisson (8), Y ~ Poisson (14) [note 2 days and hence vide (4), λ for Y = 2 x 7 = 14], Z ~ Poisson (6)

Let S = ? + ? + ? and given, ‘The malfunctioning of traffic lights between different cities is mutually independent.’, vide (5a), S ~ Poisson (28) and so

P(? + ? + ? = 24)

= e ^{– 28}.28²⁴/(24!) [vide (2)]

= 0.0601 [vide (2a)] Answer

DONE