Let f(x) = 3/64 x sqrt(16 - x^2) for 0 <= x <= 4 and f(x) =...

Let f(x) = 3/64 x sqrt(16 - x^2) for 0 <= x <= 4 and f(x) = 0 for all other values of x. (a) Verify that f is a probability density function. (b) Find P(X < 3). (Round your answer to six decimal places.)

Transcript

00:01 In this question, we're given this function, fx, for x between 0 to 4 inclusive, and fx is 0 otherwise.

00:10 In part a, you want to verify f is a probability density function.

00:14 So let's sketch the graph of fx.

00:18 So from 0 to 4, it will look something like this, and it's 0 for all other values of x.

00:30 Now, to show something is a probability density function.

00:33 We have to show the total probability is 1.

00:40 If the probability is 1, we have to find the total area under the graph because total area is actually the total probability.

00:51 So to find the total area, we just need to integrate from 0 to 4.

00:58 So let's integrate the function from 0 to 4.

01:03 Instead of square root, i'm going to write power and i'm going to use the form from f prime x x x to the power of n when we integrate this, we will get fx to the power of n plus 1 over n plus c.

01:32 So in this case, you can see that this is constant.

01:36 I can leave it outside.

01:42 This is my fx.

01:43 If i were to differentiate it, i will get minus 2x.

01:46 And you can see i have x here already.

01:49 So i must make this x minus 2.

01:51 So i'll just multiply.

01:52 A minus 2 here, so i must multiply by a minus half, so that they will balance.

02:03 So i have minus 2x, 16 minus x square to the power half dx.

02:09 So it fits the form of f prime x, and this part is my fx, and this is my n.

02:16 So it fitted this form here, so i can use the results here...

Question

Let f(x) = 3/64 x sqrt(16 - x^2) for 0 <= x <= 4 and f(x) = 0 for all other values of x. (a) Verify that f is a probability density function. (b) Find P(X < 3). (Round your answer to six decimal places.)

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