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

$P(X=x)=\dfrac{x^{119}(1-x)^{2019}}{B(120,2020)}$

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

$B(a,b)=\dfrac{\Gamma(a)\Gamma(b)}{\Gamma(a+b)}$

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

$P(X=x,Y=y)=P(X=x)P(Y=y)=\dfrac{(xy)^{119}((1-x)(1-y))^{2019}}{B(120,2020)^2}$

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

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