605-Discussion 6

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Section 4.1: Q37 - Given that P(X = a) = r, P(max(X, Y) = a) = s, and P(min(X, Y) = a) = t, show that you can determine u = P(Y = a) in terms of r, s, and t.

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Given, \(P(X = a) = r\)

\[ P\Bigg(max(X,Y)=a\Bigg)=P(X =a,Y ≤a)+P(X ≤a,Y =a)−P(X =a,Y =a) \]
\[ P\Bigg(min(X,Y)=a\Bigg)=P(X =a,Y >a)+P(X >a,Y =a)+P(X =a,Y =a). \]
Thus,

\[ => P\Bigg(max(X,Y) = a)+P(min(X,Y) = a\Bigg) = P(X = a)+P(Y = a) \]

and so,

\[ => u = t+s−r \]