Question

8. Suppose that a website receives a hit every 2 minutes on average. a) Find the probability the website receives 3 hits in 2 minutes. Show all steps and find an answer rounded off to 2 decimal places. b) Find the probability the website receives 8 or 9 hits in 10 minutes. Show all steps and find an answer rounded off to 2 decimal places. c) Find the probability the website receives more than 4 hits in 6 minutes. You do not to find the final number just indicate how you would get it.

show working please!

Answer 1

Solution Q8

Back-up Theory

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!) …………..(1)

where x = 0, 1, 2, ……. , ∞

Values of p(x) for various values of λ and x can be obtained by using Excel Function, Statistical, POISSON(x,Mean,Cumulative) ………………………………....……………………. (2)

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) ………….……….........................................….. (3)

Now to work out the solution,

Let X = Number of hits the website receives in 2 minutes. Then, X ~ Poisson (λ), where λ = 1 [given]

Part (a)

Probability the website receives 3 hits in 2 minutes

= P(X = 3)

= 0.0613 [vide (2)]

= 0.06 Answer

[By (1), the probability = e ^{– 1}.1³/(3!) = 1/6e = 1/(6 x 2.7183) = 0.0613]

Part (b)

Let Y = Number of hits the website receives in 10 minutes. Then, vide (3), Y ~ Poisson (λ),

where λ = 5 x 1 = 5 [10 = 2 x 5].

So,

Probability the website receives 8 or 9 hits in 10 minutes

= P(Y = 8 or 9)

= P(Y = 8) + P(Y = 9)

= 0.0652 + 0.0363[vide (2)]

= 0.1015

= 0.10 Answer

[By (1), the probability = e ^{– 5}[{5⁸/(8!)} + {5⁹/(9!)}]

= (e ^{– 5} x 5⁸/8!){1 + (5/9)}

= (e ^{– 5} x 5⁸ x 14)/(9!)

= (0.0067 x 390625 x 14)/362880

= 0.101544]

Part (c)

Let Z = Number of hits the website receives in 6 minutes. Then, vide (3), Z ~ Poisson (λ),

where λ = 3 x 1 = 3 [6 = 2 x 3].

So,

Probability the website receives more than 4 hits in 6 minutes

P(Z > 4)

= Σ(z = 5 to ∞){e ^{– 3}.3^x/(x!)} Answer

Or, alternatively, P(Z > 4) = 1 - P(Z ≤ 4)

= 1 - e ^{– 3}[{3⁰/(0!)} + {3¹/(1!)} +{3²/(2!)} +{3³/(3!)} +{3⁴/(4!)}

= 1 – (12.375 x e ^{– 3})

= 1 - (16.375 x 0.0498)

= 1 - 0.8153

= 0.1847

DONE