A fireworks show is designed so that the time between fireworks is between one and five seconds, and follows a uniform
distribution.
a. Find the average time between fireworks.
b. Find probability that the time between fireworks is greater than four seconds.
To find the average time between fireworks, you need to calculate the mean of the uniform distribution.
a. The average of a uniform distribution is given by the formula:
Average = (a + b) / 2,
where a and b are the minimum and maximum values of the distribution.
In this case, the minimum time between fireworks is one second (a = 1) and the maximum time is five seconds (b = 5).
Therefore, the average time between fireworks is:
Average = (1 + 5) / 2 = 6 / 2 = 3 seconds.
b. To find the probability that the time between fireworks is greater than four seconds, you need to find the area under the probability density curve of the uniform distribution for values greater than four.
The probability density function (PDF) for a uniform distribution is given by:
PDF = 1 / (b - a) = 1 / (5 - 1) = 1 / 4.
The probability that the time between fireworks is greater than four seconds is equal to the area under the probability density curve from four to five seconds.
Since the area under a uniform distribution is equal to the width of the interval, the probability in this case is:
Probability = 1 / (b - a) = 1 / 4 = 0.25 or 25%.
Therefore, the probability that the time between fireworks is greater than four seconds is 0.25 or 25%.