Problem 1 15 pts a flood control system is designed such...

Problem 1 (15 pts) A flood-control system is designed such that the probability that the system capacity is exceeded $X$ times in 30 years is given by the following discrete probability distribution: $x_i$ $f(x_i)$ $x_i$ $f(x_i)$ 0 0.04 5 0.10 1 0.14 6 0.05 2 0.23 7 0.02 3 0.24 8 0.01 4 0.18 >9 0.00 (a) What is the mean number of system failures expected in 30 years? (b) What are the variance and skewness of the number of failures?

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Transcript

00:01 Ok, so part a asks us to define the random variable x.

00:04 Now, we're basically told this in the question, and it's just the time in minutes until the next bus departs from this depot.

00:28 Part b says x and then does a little twiddle.

00:32 And usually we're writing in the distribution here.

00:36 So x is uniformly distributed from 25 to 45 minutes with constant probability to take any value in that region.

00:45 And so the graph of the distribution will be if we have x here and the probability up here, then from 25 to 45, we have a probability of 1 over 20 and it's 0 elsewhere.

01:04 Part d asks the name of the distribution.

01:06 So it's a uniform distribution and it is continuous.

01:15 Sorry.

01:15 For part e, we want the mean.

01:17 The mean of a uniform distribution is given by the lower bound plus the upper bound divided by 2, which is 35.

01:26 Part f wants the standard deviation, which is given by the upper bound minus the lower bound divided by the square root of 12.

01:34 And that turns out to be 5 .77 to two decimal places.

01:42 Part g asks the probability that it's at most 30.

01:49 So that's the probability that x is less than or equal to 30 is equal to the width of that relevant interval.

01:58 So we can draw the graph like they ask us to.

02:00 So we know it can go from 25 to 45...