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.)
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.
To find the mean of the random variable defined by the time between successes, we need to use the fact that the mean of a Poisson distribution is equal to the reciprocal of the rate parameter of the corresponding exponential distribution.
First, we need to calculate the rate parameter λ of the Poisson distribution. The given mean of 6 successes over a 126-minute period gives us:
λ = mean = 6 / 126 = 0.04762
So, the rate parameter λ of the appropriate exponential distribution is 0.04762.
The mean of the random variable, defined by the time between successes, is equal to the reciprocal of the rate parameter:
Mean = 1 / λ = 1 / 0.04762 ≈ 21
Therefore, the mean of the random variable is approximately 21.
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.
P(X > 52) = 1 - P(X ≤ 52)
The cumulative distribution function (CDF) of the exponential distribution is given by:
CDF(x) = 1 - e^(-λx)
where x is the desired time.
Using the rate parameter λ = 0.04762, we can calculate:
P(X > 52) = 1 - P(X ≤ 52)
= 1 - (1 - e^(-0.04762 × 52))
= e^(-0.04762 × 52)
≈ 0.2112
Therefore, the probability that the time to success will be more than 52 minutes is approximately 0.2112.
To answer these questions, we need to understand the relationship between the Poisson distribution and the exponential distribution.
The Poisson distribution describes the number of events that occur in a fixed interval of time or space. It is typically used when events occur at a constant average rate and independently of each other.
The exponential distribution, on the other hand, describes the time between events in a Poisson process. It is frequently used to model the time it takes for the next event to occur.
a. The mean of the random variable, defined by the time between successes in a Poisson process, can be found by taking the reciprocal of the rate (mean) of the Poisson process. In this case, the rate of the Poisson process is given as 6 successes over a 126-minute period. Therefore, the mean of the random variable is 126 / 6 = 21 minutes.
b. The rate parameter of the exponential distribution is the reciprocal of the mean of the random variable. In this case, the mean of the random variable is 21 minutes. Therefore, the rate parameter of the exponential distribution is 1 / 21 = 0.0476 (rounded to 2 decimal places).
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 the equation P(X > x) = 1 - e^(-λx), where X is the random variable, λ is the rate parameter of the exponential distribution, and x is the time period.
In this case, x is 52 minutes and the rate parameter of the exponential distribution is 0.0476. Plugging these values into the equation, we get P(X > 52) = 1 - e^(-0.0476 * 52).
To calculate this probability, we need to calculate e^(-0.0476 * 52) and subtract the result from 1.
Using a scientific calculator or programming language, we find that e^(-0.0476 * 52) ≈ 0.4231 (rounded to 4 decimal places).
Therefore, P(X > 52) = 1 - 0.4231 = 0.5769 (rounded to 4 decimal places).
So the probability that the time to success will be more than 52 minutes is approximately 0.5769.