Let x be a continuous random variable with the probality...

Let X be a continuous random variable with the probality density function f( x ) = C x, 5 ? x ? 11, zero otherwise. a. Find the value of C that would make f( x ) a valid probability density function. b. Find the probability P( X > 7 ). c. Find the mean of the probability distribution of X. d. Find the median of the probability distribution of X.

Transcript

00:01 So if i read your question correctly, it looks to me that you have some function that's a continuous random variable, a density curve, and it's cx.

00:12 And that i believe the way you typed it, i think that's supposed to be that your values go from 5 to 11.

00:19 So it looks like your function is going to look like this.

00:22 I'll kind of graph it over here.

00:25 We know that you're going to go up to, here's going to be say this is 5 and say that this is 10.

00:31 10, and this is 15.

00:34 And i'm not going to label this axis because i don't know what to go to.

00:38 But you know that you have cx, which means it's going to be some straight line that's going to look like this, going to go through the origin.

00:45 You just don't know what the slope is.

00:47 And five, if i go up to that curve, and i go up from 11, it's 11 right there, and i go up to that curve, that this is your probability distribution here.

00:59 And we know that this has to have an area underneath it for the density curve equal to 1.

01:05 And so this is the shape of a trapezoid.

01:07 And again, this line is of this equation, y equals cx.

01:13 So this point right here is the point five.

01:19 And when we plug five in here, we would get 5c, which means 5c is how long this segment is.

01:27 We just don't know what c is.

01:29 And this point right here is the point 11.

01:33 And then when we plug 11 in here, we would get 11c.

01:38 So we need this area to be one.

01:41 So let's figure out how do you find the area of a trapezoid? and for a trapezoid, we know that an area is one -half, the sum of the bases times the height, which means we will have one -half, and then the sum of these bases.

01:57 This base, so this is the two parallel sides, are right here.

02:02 So this is a base and this is also a base.

02:04 So this is 5c long and this base is 11c long.

02:15 And then the height of this trapezoid is this length right here.

02:20 And that goes from 11, 5 to 11.

02:22 So that difference is 6.

02:23 And we need that area to equal one.

02:28 So we just need to solve this equation.

02:30 So when half of six is three, so i have 16c and then times three has to equal one, therefore we know that 48c has to equal one.

02:47 So c has to be one 48th.

02:50 So the function is actually 148x, and that goes from 5 up to 11.

03:03 And then we wanted to find what was the likelihood.

03:08 I think it said something about what was the probability that you randomly pick a number, and the number was greater than 7 for this distribution.

03:16 So let me make our little depiction here.

03:19 So let's say 7 is about here.

03:22 So we really need to find this area.

03:27 And this point is the point seven.

03:31 And when we plug seven into this function, we get 7 over 48.

03:37 So that's how long this segment is.

03:40 So we need to find that area.

03:42 I'll continue doing the area in red.

03:44 So i know it's a trapezoid.

03:46 I have one half the sum of the bases.

03:49 So this base is 7 .8s.

03:51 Excuse 748s.