suppose $X$ and $Y$ are independent random variables with standard normal distributions. The probability of $X<-1$ is some p which lies in the open interval (0,1). what is the probability of the event: $X^2 > 1$ and $Y^3 > 1$.
I have started with $P(X<-1)= P(X>1) = p$ (i think its right)
But now stuck at how to proceed from here… I'm thinking of using cdf but not able to fit it in?
Best Answer
$$P((X^2 > 1) \cap Y^3 > 1)$$ $$=P(X^2 > 1)\times P(Y > 1)$$ $$=P(X > 1)\times P(Y>1) + P(X < -1)\times P(Y>1)$$
$$=(1-\Phi(1))(1-\Phi(1)) + \Phi(-1)\times (1-\Phi(1))$$
where $\Phi(.)$ denotes the CDF of $N(0,1)$