a. The mean of the random variable, defined by the time between successes, is equal to the reciprocal of the mean of the original Poisson random variable. Therefore, the mean of the time between successes is 1/6.
b. The rate parameter of the appropriate exponential distribution is equal to the mean of the time between successes, which is 1/6.
c. To find the probability that the time to success will be more than 52 minutes, we can use the cumulative distribution function (CDF) of the exponential distribution.
The CDF of the exponential distribution is given by: P(X > x) = 1 - e^(-λx), where λ is the rate parameter and x is the desired time.
In this case, λ = 1/6 and x = 52 minutes. Substituting these values into the equation: P(X > 52) = 1 - e^(-(1/6)*52).
Calculating this, we find: P(X > 52) ≈ 0.2814.
Therefore, the probability that the time to success will be more than 52 minutes is approximately 0.2814.
Assume a Poisson random variable has a mean of 6 successes over a 126-minute period.
a. Find the mean of the random variable, defined by the time between successes.
b. What is the rate parameter of the appropriate exponential distribution? (Round your answer to 2 decimal places.)
c. Find the probability that the time to success will be more than 52 minutes. (Round intermediate calculations to 2 decimal places. Round your final answer to 4 decimal places.)
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