Question

Let X and Y be two independent random variables following beta distributions Beta(120, 2020).

1. What's P(X = 0.5)?
2. What's P(X + 3 < 2Y)?
3. What's P(X > Y)?

Answer 1

With
`X,Y\sim\mathrm{Beta}(120,2020)`, we have identical PDFs

`P(X=x)=(x^(119)(1-x)^(2019))/(B(120,2020))`

for
`0<x<1`, and 0 otherwise, where

`B(a,b)=(\Gamma(a)\Gamma(b))/(\Gamma(a+b))`

Since
`X,Y` are independent, the joint PDF is

`P(X=x,Y=y)=P(X=x)P(Y=y)=((xy)^(119)((1-x)(1-y))^(2019))/(B(120,2020)^2)`

for points
`(x,y)` in the unit square, and 0 otherwise.

1. The distribution is continuous, so
`P(X=0.5)=\boxed0`.

2.
`X+3<2Y` is the region in the
`x,y` plane contained within the unit square and above the line
`y=\frac{x+3}2`. This region is empty, because this line lies above the square altogether, so
`P(X+3<2Y)=\boxed0`.

3.
`X>Y` is the region in the same square below the line
`y=x`. So we have

`P(X>Y)=\displaystyle\int_0^1\int_0^xP(X=x,Y=y)\,\mathrm dy\,\mathrm dx=\boxed{\frac12}`