Let x be a continuous random variable with probability...

Let X be a continuous random variable with probability density function given by f(x)= -(1/2)x + 1, 0 ≤ x ≤ 2. (a) Graph the density function f (x). b) Find the total area under f (x) for 0 ≤ x ≤ 2. c) Find P(X≥1) using geometry and integration. d) Find P(X≤1/2). e) Find P(X = 1½). f) Find the expected value and variance of X. g) Find the cumulative distribution function of X.

Transcript

00:01 So let's work through each part of this problem step by step.

00:08 Number one, graph the density function f of x.

00:12 So the density function f of x is given as negative one half x plus one four zero to x to two.

00:30 This is a linear function.

00:33 To graph it, plot the points zero one and two zero and connect them with a straight line.

00:42 Here's a rough sketch.

00:47 So it would be zero one and two and then zero and one.

00:53 So you plot zero one and two zero and kind of align between there.

01:12 B, so this is actually a.

01:15 B, find the total area under f of x for x between zero and two.

01:21 So we need to find the integration of zero to two f of x dx.

01:34 Dx equals the integration of zero to two negative one half x plus one and that's dx.

02:01 So we need to integrate with respect to x.

02:06 So that would be negative one fourth x squared plus x from zero to two.

02:20 Integrate and we get one.

02:24 So that's the total area under the curve f of x for x between zero and two.

02:32 C, find p of x greater than or equal to one using geometry and integration.

02:43 So p of x greater than or equal to one represents the area under the curve f of x to the right of x equals one.

02:51 This can be found by calculating the definite integral from one to two.

02:55 So this would equal integral one to two dx which comes out to be one fourth...

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