Chi-squared variables CDF

Question

I have a random variable $Y=X^2_1+X^2_2+X^2_3$ where $X_i$ are i.i.d. random variables with a standard normal distribution. I have to calculate the CDF $P(Y\le7,8)$. I try to calculate $P(3X^2\le 7,8)$ but the result is wrong. How can i solve it?

Answer 1

The calculation $P(3X^2\le7.8)$ assumes all the $X_i$ are identical in values, whereas they are in fact only iid.

You need a dedicated $\chi^2$ table for finding $P(Y\le7.8)=P(\chi^2(3)\le7.8)$, and here the result is around $0.95$.