Question

Let ( X ) and ( Y ) be two discrete RVs: [ Pleft{X=x_{1} ight}=p_{1}, quad Pleft{X=x_{2} ight}=1-p_{1}, ] and [ Pleft{Y=y_{1} ight}=p_{2}, quad Pleft{Y=y_{2} ight}=1-p_{2} . ] Show that ( X ) and ( Y ) are independent if and only if the covariance between ( X ) and ( Y ) is zero.

          Let ( X ) and ( Y ) be two discrete RVs:
[
Pleft{X=x_{1}
ight}=p_{1}, quad Pleft{X=x_{2}
ight}=1-p_{1},
]
and
[
Pleft{Y=y_{1}
ight}=p_{2}, quad Pleft{Y=y_{2}
ight}=1-p_{2} .
]

Show that ( X ) and ( Y ) are independent if and only if the covariance between ( X ) and ( Y ) is zero.

Let ( X ) and ( Y ) be two discrete RVs:
[
PleftX=x1
ight=p1, quad PleftX=x2
ight=1-p1,
]
and
[
PleftY=y1
ight=p2, quad PleftY=y2
ight=1-p2 .
]

Show that ( X ) and ( Y ) are independent if and only if the covariance between ( X ) and ( Y ) is zero.

Added by Jaki B.

Probability with Applications in Engineering, Science, and Technology

Matthew A. Carlton • Jay L. Devore 2nd Edition

Chapter 4

Instant Answer

Solved by Expert Shu Naito

04/15/2024

Step 1

First, recall the definition of covariance: \[Cov(X, Y) = E[XY] - E[X]E[Y]\] If X and Y are independent, then by definition, \(E[XY] = E[X]E[Y]\). Show more…

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