In Section 5.2, we argued that if Y1 and Y2 have joint

QUESTION:

In Section 5.2, we argued that if \(Y_{1}\) and \(Y_{2}\) have joint cumulative distribution function \(F\left(y_1,\ y_2\right)\) then for any \(a<b\) and \(c<d\)

                         \(P\left(a<Y_1\le b,\ c<Y_2\le d\right)=F(b,\ d)-F(b,\ c)-F(a,\ d)+F(a,\ c).\)

If \(Y_{1}\) and \(Y_{2}\) are independent, show that

                         \(P\left(a<Y_1\le b,\ c<Y_2\le d\right)=P\left(a<Y_1\le b\right)\times P\left(c<Y_2\le d\right).\)

[Hint: Express \(P\left(a<Y_{1} \leq b\right)\) in terms of \(F_{1}(\cdot)\).]

Equation Transcription:

Text Transcription:

Y_1

Y_2

F(y_1,y_2)

a<b

c<d

P(a<Y_1</=b,c<Y_2</=d)=F(b,d)-F(b,c)-F(a,d)+F(a,c).

Y_1

Y_2

P(a<Y_1</=b,c<Y_2</=d)=P(a<Y_1</=b)P(c<Y_2</=d).

P(a<Y_1</=b)

F_1(cdot)