A point is chosen at random inside a triangle with vertices at (0, 0), (0, 8), and (8, 0). The continuous random variable denotes the x-coordinate of that point. a) Find the probability that is less than 5.

b) Find the cumulative distribution function (i. e. P( ≤ x) ).

c) Find the probability density function.

d) Find the average value of X.

Assuming all points in the triangle are uniformly distributed, we have the joint density

where is the area of the triangle .

(a)

(where is the portion of for which is between 0 and 5)

(b) Generalizing the previous result, we have

(this time with being the portion of where for some between 0 and 8)

That is, the CDF of is

or

(c) Obtain the PDF by differentiating the CDF:

(d) Compute the expectation of :

answer: on graphs, the open and closed circles, or vertical asymptotes drawn as dashed lines us identify discontinuities.

as before, graphs and tables allow us to estimate at best.

when working with formulas, getting zero in the denominator indicates a point of discontinuity.

when working with piecewise-defined functions, check for discontinuities at the transition points where one piece ends and the next begins.

step-by-step explanation: