# Statistics

Suppose that X , Y , and Z are independent, with E[X]=E[Y]=E[Z]=2 , and E[X2]=E[Y2]=E[Z2]=5 .

Find cov(XY,XZ) .

cov(XY,XZ)=

Let X be a standard normal random variable. Another random variable is determined as follows. We flip a fair coin (independent from X ). In case of Heads, we let Y=X . In case of Tails, we let Y=−X .

YES
NO
not enough information to determine

Find P(X+Y≤0) .

P(X+Y≤0)=

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1. cov(XY,XZ)= 0

Is Y normal? => NO

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2. Cov(XY,XZ) is 4.

Is Y normal? --> YES
Cov (X,Y) = 0
Are X and Y independent? --> YES, because they are normally distributed and their COV is 0.

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3. Any answers in the last question?

Find P(X+Y≤0) .

P(X+Y≤0)=

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4. cov(XY, XZ) = 4

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5. Is Y normal? = Yes

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6. Cov(X,Y) = 0

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7. Are XandY independent? = NO

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8. Find P(X+Y≤0) .

P(X+Y≤0)= 3/4

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9. Not independent
Cov(x,y) is 0
P(X+Y≤0)=1/2

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