The random variable x is the number of occurrences of an event over an interval of ten minutes. it can be assumed that the probability of an occurrence is the same in any two-time periods of an equal length. it is known that the mean number of occurrences in ten minutes is 5.3. the appropriate probability distribution for the random variable

A Poisson distribution, with λ=5.3

Step-by-step explanation:

For this type of random variables, a Poisson distribution is the most adecuate.

If the number of occurrences in ten minutes is 5.3, we can define the parameter λ=5.3 and the probability function as:

a) We have the mean number of events is interval(5.3 events in 10 minutes), which constitutes the Poisson probability distribution.

b) 5.3

c) 5.3

Step-by-step explanation:

In a Poisson distribution, the probability that X represents the number of successes of a random variable is given by the following formula:

In which

x is the number of sucesses

e = 2.71828 is the Euler number

is the mean in the given time interval.

a. What probability distributions' properties is this?

We have the mean number of events is interval(5.3 events in 10 minutes), which constitutes the Poisson probability distribution.

b. What is the expected value of the random variable x?

It is known that the mean number of occurrences in ten minutes is 5.3.

So the mean is

c. What is the variance of the random variable x?

In the Poisson probability distribution, the variance is the same as the mean. So the variance is 5.3.

b. 0.0771

Step-by-step explanation:

P(X=8)=e^-535.3^8/8!

=0.0771

The probability that occurrence of 8 in ten minutes is 0.0771.

Step-by-step explanation:

Poisson Distribution:

A discrete random variable X having the enumerable set {0,1,2,.....} as the spectrum, is said to have Poisson distribution with parameter (>0), if the p.m.f is given by

 for x=0,1,2,...

               = 0,           elsewhere

The mean number of occurrences in ten minutes is 5.3.

Here = 5.3 and x= 8

               =0.0771

The probability that occurrence of 8 in ten minutes is 0.0771.

The probablility that there are 8 occurrences in ten minutes is 0.00518 or 0.5%.

Step-by-step explanation:

The distribution that represents the ocurrence of this events is the Poisson distribution.

The probability of having k events in a ten-minute-period can be expressed as:

being λ the mean number of occurrences in ten minutes.

The probablility that there are 8 occurrences in ten minutes can be calculated as:

The probablility that there are 8 occurrences in ten minutes is 0.00518 or 0.5%.