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The correct answer is C

p(X=1|Y=2)=0.05/(0.05+0.10+0.15)=0.17.

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The variance of X is closest to:

A) 0.54.

B) 0.67.

C) 0.61.

D) 0.74.

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2、Use the following joint probability distribution to answer the questions below.

	Y=1	Y=2	Y=3
X=1	0.05	0.05	0.10
X=2	0.05	0.10	0.15
X=3	0.15	0.15	0.20

The expected value of Y is closest to:

A) 2.2.

B) 0.2.

C) 1.0.

D) 2.3.

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The correct answer is A

p(Y=1)=0.05+0.05+0.15=0.25, p(Y=2)=0.05+0.10+0.15=0.30, and p(Y=3)=0.10+0.15+0.20=0.45, so E(Y)=0.25(1)+0.30(2)+0.45(3)=2.20.

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AIM 5: List and discuss the properties of variance.

1、Use the following probability distribution to calculate the standard deviation for the portfolio.

State of the Economy	Probability	Return on Portfolio
Boom	0.30	15%
Bust	0.70	3%

A)  6.0%.

B)  6.5%.

C)  7.0%.

D)  5.5%.

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The correct answer is D

[0.30 × (0.15 ? 0.066)2 + 0.70 × (0.03 ? 0.066)2]1/2 = 5.5%.

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The correct answer is D

p(Y = 1) = 0.05 + 0.15 = 0.20, p(Y = 2) = 0.15 + 0.15 = 0.30, and p(Y = 3) = 0.20 + 0.30 = 0.50. Thus, the mean of Y is equal to μ = 0.20(1) + 0.30(2) + 0.50(3) = 2.3, and the variance is calculated as 0.20(1 – 2.3)2 + 0.30(2 – 2.3)2 + 0.50(3 – 2.3)2 = 0.61.

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4、The characteristic function of the product of independent random variables is equal to the:

A) square root of the product of the individual characteristic functions.

B) exponential root of the product of the individual characteristic functions.

C) product of the individual characteristic functions.

D) sum of the individual characteristic functions.

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The correct answer is C

The characteristic function of the sum of independent random variables is equal to the product of the individual characteristic functions. E(XY) = E(X) × E(Y).

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The correct answer is A

p(X = 1) = 0.05 + 0.15 + 0.20 = 0.40, and p(X = 2) = 0.15 + 0.15 + 0.30 = 0.60, so E(X) = 0.40(1) + 0.60(2) = 1.6.