Choose a point uniformly at random in that half of the unit disk that is to the right of the y-axis. In other words, choose a point uniformly at random from the set {(x, y) : x > 0, x^2 + y^2 < 1}. Let S be the slope of the line through the chosen point and the origin. a) Find the cumulative distribution function of S. b) Find the probability density function of S.

Call the set of points (for half-disk), where in polar coordinates the region is

has area , and so points in are uniformly distributed according to the joint density,

If is some point in , then the slope of the line through and the origin is

Notice that means we can have .

a. The CDF of is then

Compute the marginal density for by integrating the joint density over the support of :

Then the marginal CDF for is

It follows that the CDF for is then

where .

b. Differentiate the CDF to find the density:

describing distributions is one of the key skills you’ll need to earn a high score on the ap statistics exam. if you need proof of this, just flip through some past exam questions, which can be found at the collegeboard website. you’ll notice that the first free response question is almost always a question requiring you to look at a graph and describe it.

step-by-step explanation: just trying to get points sorry : (

step-by-step explanation:

mean or average is 40.6