I really strugger with this question.
Two archers with probabilities p = 0.52 and q = 0.58 independantly of each other shooting at the target. Both shot n=5 times. Let's say that first archer hitted X times while second archer hitter Y times. Find Z = max(X,Y) value of the distribution function at x = a(a=2.51).
I don't understand how to calculate Z = max(X,Y) part.
Any help appreciate, thanks.
We can compute the distribution function of $Z$ using the fact that $$ Z\leq z\iff X\leq z \; \text{and}\; Y\leq z $$ In particular $$ P(Z\leq z)=P(X\leq z) P(Y\leq z) $$ by independence. We want $$ P(Z\leq 2.51)=P(Z\leq 2)=P(X\leq 2) P(Y\leq 2) $$ since $Z$ is integer valued. Since $X$ and $Y$ are binomially distributed the probabilities on the right hand side are easy to compute.