Question

There are 12 accidents per month on a highway. Define the random variable X= no.of accidents in a week.

a) What distribution does X follow?

b) Find the mean, variance, standard deviation of the random variable X.

c) Find the probability of producing 5 accidents in a week.

Answer 1

Answer 2

a) X follows a Poisson distribution, since it is a count of the number of accidents occurring in a fixed interval of time, given a known rate.

b) The mean of a Poisson distribution is equal to the rate parameter, which is given as 12 accidents per month. To find the rate per week, we divide by 4, since there are four weeks in a month:

Mean (μ) = rate/week = 12/4 = 3 accidents/week

The variance of a Poisson distribution is also equal to the rate parameter, so:

Variance (σ^2) = rate/week = 3 accidents/week

The standard deviation is the square root of the variance:

Standard deviation (σ) = sqrt(3) ≈ 1.732

c) The probability of producing 5 accidents in a week can be calculated using the Poisson distribution formula:

P(X = 5) = (e^(-μ) * μ^x) / x!

where μ is the mean and x is the number of accidents we want to find the probability for.

Plugging in the values:

P(X = 5) = (e^(-3) * 3^5) / 5!

P(X = 5) = 0.1008 (rounded to four decimal places)

Therefore, the probability of producing 5 accidents in a week is approximately 0.1008 or 10.08%.