Let XXX and YYY be independent with X∼Gamma(α,1)X \sim \text{Gamma}(\alpha, 1)X∼Gamma(α,1) and Y∼Gamma(β,1)Y \sim \text{Gamma}(\beta, 1)Y∼Gamma(β,1). What is P(X<Y)P(X < Y)P(X<Y)?
αα+β\frac{\alpha}{\alpha + \beta}α+βα
βα+β\frac{\beta}{\alpha + \beta}α+ββ
0.50.50.5
Depends on specific values