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A box contains three cards, labeled 1, 2, and 3. Two cards are chosen at random, with the first card being replaced before the second card is drawn. Let X represent the number on the first card, and let Y represent the number on the second card.
a. Find the joint probability mass function of X and Y.
b. Find the marginal probability mass functions pX(x)and pY(y).
c. Find μX and μY.
d. Find μXY.
e. Find Cov(X, Y).

Sagot :

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Answer:

___Y/X : ___ 1 ___ 2 ____ 3 __total

____1 _____1/9 __ 1/9 ___1/9 __ 1/3

____2_____1/9 __ 1/9 ___1/9 __ 1/3

____3_____1/9 __ 1/9 ___1/9 __ 1/3

___Total __ 1/3 __ 1/3 ___ 1/3 __ 1

Y: ____ 1 _____ 2 _____ 3

P(Y) : _ 1/3 ____ 1/3 ____ 1/3

μXY = E(X) * E(Y) = 2 * 2 = 4

Cov(X, Y) = 0

Step-by-step explanation:

Sample space = 3² = 9

___Y/X : ___ 1 ___ 2 ____ 3 __total

____1 _____1/9 __ 1/9 ___1/9 __ 1/3

____2_____1/9 __ 1/9 ___1/9 __ 1/3

____3_____1/9 __ 1/9 ___1/9 __ 1/3

___Total __ 1/3 __ 1/3 ___ 1/3 __ 1

The probability density function :

X : __ 1 ____ 2 ____ 3

P(X) : 1/3 ___1/3 ____ 1/3

μX =E(X) = ΣX*P(X) = (1*1/3) + (2*1/3) * (3*1/3)

= 1/3 + 2/3 + 1 = (1 + 2 + 3) / 3 = 6/3 = 2

Y: ____ 1 _____ 2 _____ 3

P(Y) : _ 1/3 ____ 1/3 ____ 1/3

μY =E(Y) = ΣY*P(Y) = (1*1/3) + (2*1/3) * (3*1/3)

= 1/3 + 2/3 + 1 = (1 + 2 + 3) / 3 = 6/3 = 2

μXY = E(X) * E(Y) = 2 * 2 = 4

Cov(X, Y) = 0

X and Y are independent

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