Based on an analysis of sample data an article proposed the...

Based on an analysis of sample data, an article proposed the pdf f(x) = 0.25e^(-0.25(x - 1)) when x ≥ 1 as a model for the distribution of X = time (sec) spent at the median line. (Round your answers to three decimal places.) (a) What is the probability that waiting time is at most 6 sec? More than 6 sec? P(X ≤ 6) = P(X > 6) = (b) What is the probability that waiting time is between 4 and 9 sec?

Transcript

00:01 So we're told the probability of the waiting time in the median, it's given as the pdf of 0 .25 times e to the negative 0 .25 times x minus 1.

00:14 And this is for x greater than equal to 1 and it's 0 otherwise.

00:20 And so what we're going to do for part a is find the probability that the waiting time is at most 6 seconds.

00:27 So the probability x is less than re -fold to 6.

00:31 You want to find the probability that x is more than six.

00:41 So we want to find those two probabilities.

00:43 And we don't need to worry about the less than equal two versus the greater than because the probability of any specific value is zero.

00:51 So you don't really need to worry about this in a continuous distribution because this is, we don't need to worry about that.

00:58 But the direction we do very much need to worry about.

01:00 So the first one is going to be evaluated.

01:04 It's the integral from 0 to 6 of the function f of x integrated with respect to x.

01:12 And this one being more than 6, well, we'll say 1 minus.

01:17 Well, we could do a few things.

01:17 We could do from 6 up to infinity, f of x, dx.

01:23 We could do that.

01:24 But we're already going to have 0 to 6.

01:27 So what we can do is 1 minus the integral from 0 to 6.

01:31 Of f of x d x because they're complementing events.

01:35 So let's go ahead and do that.

01:36 We'll find the integral of f of x from 0 to 6.

01:43 So we'll factor up the 0 .25 because it's a constant.

01:47 So we have e to the negative 0 .25 times x.

01:51 But i'm going to distribute the negative 0 .25.

01:56 So we have that plus 0 .25.

01:59 Right, because you negative 0 .25 times second of 1 is positive 0 .25.

02:06 X and let's see this we can since we're adding these exponents we can actually think about this as e to the 0 .25 times e to the negative 0 .25 x which means this factors are constant we can bring it out so you get 0 .25 e to 0 .25 0 to 6 of e to negative 0 .25 x d x and this is a lovely thing to integrate the function e, boy, it's just beautiful.

02:42 So we get 0 .25e to the 0 .25.

02:51 And we get e to the negative 0 .25x divided by negative 0 .25.

03:00 I always, you know, i always forget to bring in the negative when you integrate this.

03:04 So just be aware of that.

03:04 You got to be able to do that.

03:06 Zero to six.

03:09 So let's see...