Question

Assume a Poisson random variable has a mean of 6 successes over a 120-minute period a. Find the mean of the random variable, defined by the time between successes Mean b. What is the rate parameter of the appropriate exponential distribution? (Round your answer to 2 decimal places.) Rate parameter C. Find the probability that the time to success will be more than 60 minutes. (Round intermediate calculations and final answer to 4 decimal places.) Probability

Answer 1

Answer:

a) The mean is 20 minutes per success.

b) The rate parameter of the exponential distribution is  λ=0.05 successes/min.

c) The probability that the time to success will be more than 60 minutes is 0.0498.

Step-by-step explanation:

a) The mean of the random variable T, time between successes can be calculated as:

$T=\frac{120\,min}{6 \,suc} =20\,min/successes$

The mean is 20 minutes per success.

b) The parameter λ can be expressed as the rate of successes over time:

$\lambda=\frac{6}{120} =0.05$

The rate parameter of the exponential distribution is  λ=0.05 successes/min.

c) The probability that the time to success will be more than 60 minutes can be expressed as:

$P(X>60)=1-P(X<60)=1-(1-e^{-\lambda t})\\\\P(X>60)=1-1+e^{-0.05*60}=e^{-3}=0.0498$

The probability that the time to success will be more than 60 minutes is 0.0498.