Q: If $$E(X) = -0.05263$$ for a $$1 bet on red, what is the expected value for a 5$ bet on red?

$$E(5X) = 5 \times E(X) = -0.2632$$

Question 1

Which of the following best describes a discrete random variable?

Accepted Answer

A variable whose possible values are countable and distinct, such as 1, 2, 3, etc.

Question 2

Given the probability model for points scored on a red zone possession: $$f(x_0$) = 12/46$$ when $$x=0$$, $$10/46$$ when $$x=3$$, $$24/46$$ when $$x=7$$, and $$0$$ otherwise, what is the expected value $$E(X)$$?

Accepted Answer

$$E(X) = 4.304$$

Question 3

Which formula correctly represents the variance of a discrete random variable $$X$$ with mean $$\mu$$?

Accepted Answer

$$Var(X) = \sum_{i=1}^{N} (x_i - \mu)^2$ \cdot f(x_i)$$

Question 4

A $$1 bet on red in Roulette has f$$($$x_0) = 18/38$ for x=+1$ and 20/38$ for x=-1$. What is the expected return E(X$$)$?

Accepted Answer

$$E(X) = -0.05263$$

Question 5

Which of the following statements about expected value is TRUE?

Accepted Answer

Expected value is a measure of central tendency for a random variable.

Question 6

If $$E(X) = -0.05263$$ for a $$1 bet on red, what is the expected value for a 5$ bet on red?

Accepted Answer

$$E(5X) = 5 	imes E(X) = -0.2632$$

Question 7

Which property of variance is correct for a constant $$a$$ and random variable $$X$$?

Accepted Answer

$$Var(aX) = a^2$ Var(X)$$

Question 8

If $$Var(X) = 0.9972$$ for a $$1$$ bet on red, what is the standard deviation for a $$5$$ bet on red?

Accepted Answer

$$SD(5X) = 5 	imes SD(X)$$

Question 9

Which of the following is a correct property of expected values for independent random variables $$X_1$$ and $$X_2$$?

Accepted Answer

$$E(X_1$ + X_2$) = E(X_1$) + E(X_2$)$$

Question 10

If $$Var(X_1$) = 2.9167$$ and $$Var(X_2$) = 2.9167$$ for two independent dice, what is $$Var(X_1$ + X_2$)$$?

Accepted Answer

$$Var(X_1$ + X_2$) = 5.833$$

Question 11

Which of the following is NOT a property of expected value?

Accepted Answer

$$E(X^2$$) = [E(X)]^2$$

Question 12

A die is rolled. What is the expected value of the number facing up?

Accepted Answer

$$E(X) = 3.5$$

Question 13

If $$X_1$$ and $$X_2$$ are independent, what is $$Var(X_1$ - X_2$)$$?

Accepted Answer

$$Var(X_1$ - X_2$) = Var(X_1$) + Var(X_2$)$$

Question 14

Which of the following is TRUE about the variance of a $$1$$ bet on red versus a $$1$$ straight bet on a single number in Roulette?

Accepted Answer

The variance of a straight bet is much larger than the variance of a bet on red.

Question 15

If $$Var(X_1$) = 2.9167$$ and $$Var(X_2$) = 2.9167$$, what is the standard deviation of $$X_1$$ + $$X_2$$?

Accepted Answer

$$SD(X_1$ + X_2$) = \sqrt{5.833}$$

Question 16

Which of the following is a correct statement about standard deviations?

Accepted Answer

Standard deviations cannot be added directly for independent random variables.

Question 17

If $$X is a $$random variable and $$a is a $$constant, which of the following is TRUE?

Accepted Answer

$$SD(aX) = |a| SD(X)$$

Question 18

Which of the following is the correct formula for the expected value of a discrete random variable $$X$$?

Accepted Answer

$$E(X) = \sum_{i=1}^{n} x_i \cdot f(x_i)$$

Random Variables, Expected Value, Variance, and Standard Deviation in Business Statistics

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