Cumulative distribution function

[magnifik]

A dart is thrown towards a quadrilateral defined by {(x,y): 0 < x < b, 0 < x < b}. Assume the dart is equally likely to land anywhere within this shape. Let Z be denoted by the (x,y) coordinate with the least value. Find the region in the square corresponding to {Z < z}

so i know the sample space contains any x or y within 0 to b in both directions. and i know that Z = whichever coordinate is the smallest, with a minimum at 0 and a maximum at b. i don't understand how to find where {Z < z} inside the square... wouldn't all the values be within Z? this makes no sense to me

 

[Ray Vickson]

A dart is thrown towards a quadrilateral defined by {(x,y): 0 < x < b, 0 < x < b}. Assume the dart is equally likely to land anywhere within this shape. Let Z be denoted by the (x,y) coordinate with the least value. Find the region in the square corresponding to {Z < z}

so i know the sample space contains any x or y within 0 to b in both directions. and i know that Z = whichever coordinate is the smallest, with a minimum at 0 and a maximum at b. i don't understand how to find where {Z < z} inside the square... wouldn't all the values be within Z? this makes no sense to me

If I give you a value of z in (0,b) it is quite possible for both X and Y to both take values > z, so Z (the smaller of X and Y) would be > z in that case. By the way, you should distinguish between X (the random variable) and x (possible numerical value for X), and between Y and y.

RGV

 

[magnifik]

i'm still confused on how this relates to the area of the square

 

[magnifik]

is z the entire area of the square since it can take on any value from 0 to b in both the x and y directions?