Question

The number of motor-vehicle fatalities in a city during a week is Poissondistributed, with an average of 8 fatalities per week.

a) What is the probability that 12 fatalities occur in a specific week?
b) What is the probability that at least 12 fatalities occur in a specific week?
c) How many motor-vehicle fatalities would have to occur during a given week to conclude there are an unusually high number of events in that week?

Answer 1

In Poisson distribution problems, we calculate probabilities using the formula (e^-λ * λ^x) / x!. We can find the probability of a specific number of events occurring and determine if the observed number of events is unusual.

Step-by-step explanation:

To calculate the probabilities in a Poisson distribution problem, we can use the formula:

P(x) = (e-λ * λx) / x!

Where P(x) represents the probability of x events occurring, λ is the average number of events per time period, and x is the number of events we want to calculate the probability for.

a) To find the probability that 12 fatalities occur in a specific week with an average of 8 fatalities per week, we substitute λ = 8 and x = 12 into the formula:

P(12) = (e-8 * 812) / 12!

b) To find the probability that at least 12 fatalities occur in a specific week, we need to calculate the sum of the probabilities for 12,13,14,... until infinity.

c) To determine an unusually high number of events, we can compare the observed number of events to the expected number of events based on the average rate. If the observed number is significantly higher than the expected number, it can be considered unusual.

Answer 2

Answer: a) 0.0481, b) 0.921, c) 13

Step-by-step explanation:

Since we have given that

`\mu=8`

We will use Poisson distribution:

a) What is the probability that 12 fatalities occur in a specific week?

`P(X=x)=(e^(-\lambda)* \lambda^x)/(x!)\\\\P(X=12)=(e^(-8)* 8^(12))/(12!)\\\\P(X=12)=0.0481`

b) What is the probability that at least 12 fatalities occur in a specific week?

`P(X\leq 12)=P((X-\lambda)/(√(\lambda))\leq (12-\lambda)/(√(\lambda)))\\\\P(X\leq 12)=P(z\leq (12-8)/(√(8)))\\\\P(X\leq 12)=P(z\leq 1.4142)\\\\P(X\leq 12)=0.921`

c) How many motor-vehicle fatalities would have to occur during a given week to conclude there are an unusually high number of events in that week?

`(x_0-\lambda)/(√(\lambda))=1.645\\\\(x_0-8)/(√(8))=1.645\\\\x_0-8=1.645* √(8)=4.652\\\\x=4.652+8\\\\x=12.652\\\\x\approx 13`

Hence, a) 0.0481, b) 0.921, c) 13