JAM 2022 [ 41 -50]

Let X1 and X2 be two independent and identically distributed random variables having \chi_2^2 distribution and W = X1 + X2. Then P(W> E(W)) equals _____________ (round off to 2 decimal places)

Since X1 and X2 are two independent and identically distributed random variables having chi-square distribution with degrees of freedom 2 each. W = X1 + X2 will follow a chi-square distribution with degrees of freedom 4. By property of chi-square distributions,

E(W)=4 , V(W)=8 \\
P(W>E(W)) = P(W>4) = \frac{1}{4} \int_{4}^{\infty} y^{2-1} e^{-y/2} dy \\
=  \int_{2}^{\infty} y e^{-y} dy = \left[ -ye^{-y} - e^{-y} \right]_{2}^{\infty} \\
=3e^{-2}=0.406