I have a problem in numerically evaluating the PDF of $Y=\min(X_1,X_2,\cdots,X_N)$ where $N=\binom{M}{K}$, the binomial coefficient and $X_i$s are iid Chi-square random variables.
The CDF of $Y$ is $F_Y(y)=1-[1-F_X (y)]^N$, where $F_X (y)$ is the CDF of an $X_i$.
Then the PDF of $Y$ is $P_Y (y)=\frac{d}{dy}F_Y (y)=N [1-F_X (y)]^{N-1}P_X (y)$.
I have tired to plot $P_Y (y)$ for $M=100$ and $K=10$, I do not have any problem. When $K$ increases keeping $M$ as constant, the PDF of $P_Y (y)$ exhibit a sharp transition from 0 to 1 in the lower tail and the area under the PDF does not equal to 1 numerically. However, analytically the area can be shown to be equal to 1, that is, $\int_{0}^{\infty} P_Y (y) dy=1.$
My questions are
- Why does the area does not equal to one when N increases? Is there a discontinuity happening as N increases?
We note that $0 \leq F_X (y) \leq 1$. Then, the term $[1-F_X (y)]^{N-1}$ in $P_Y (y)$ for large $N$ becomes a binary threshold function. Is there is a problem with the binary threshold function such as discontinuity?
Please help me, I do not know where I am making mistake.