Q: Suppose a company finds that 60% of its customers renew their annual subscription. If 10 customers are randomly selected, what is the expected number of renewals?

$$E(X) = np = 10 \times 0.6 = 6$$

Q: A binomial random variable $$X$$ has parameters $$n = 8$$ and $$p = 0.25. $$What is the standard deviation of $$X$$?

$$SD(X) = \sqrt{np(1-p)} = \sqrt{8 \times 0.25 \times 0.75} \approx 1.225$$

Q: A casino models its high-roller losses as $$N(\mu = 850,000, \sigma = 60,000). $$What loss threshold should the casino set to only offer discounts to the top 1% of losers?

$$x^* = \mu + z_{0.99} \sigma = 850,000 + 2.326 \times 60,000 \approx 989,560$$

Q: A portfolio consists of 30% stocks ($$\mu_S = 0.115$$, $$\sigma_S = 0.199) $$and 70% bonds ($$\mu_B = 0.044$$, $$\sigma_B = 0.046$$), with returns assumed independent and normal. What is the mean and standard deviation of the portfolio return?

$$\mu_P = 0.3 \times 0.115 + 0.7 \times 0.044 = 0.0653$$, $$\sigma_P = \sqrt{(0.3^2$ \times 0.199^2$) + (0.7^2$ \times 0.046^2$)} \approx 0.0678$$

Q: A normal random variable $$X$$ has $$\mu = 10$$ and $$\sigma = 2. $$What value $$x^*$$ satisfies $$P(X \u003c x^*) = 0.975$$?

$$x^* = 10 + 1.96 \times 2 = 13.92$$

Question 1

Which of the following is NOT a property of a Bernoulli trial?

Accepted Answer

There are more than two possible outcomes.

Question 2

Suppose a company finds that 60% of its customers renew their annual subscription. If 10 customers are randomly selected, what is the expected number of renewals?

Accepted Answer

$$E(X) = np = 10 	imes 0.6 = 6$$

Question 3

A binomial random variable $$X$$ has parameters $$n = 8$$ and $$p = 0.25. $$What is the standard deviation of $$X$$?

Accepted Answer

$$SD(X) = \sqrt{np(1-p)} = \sqrt{8 	imes 0.25 	imes 0.75} \approx 1.225$$

Question 4

Which scenario is best modeled by a binomial distribution?

Accepted Answer

Counting the number of defective products in a batch of 100, where each product has a 2% chance of being defective.

Question 5

If $$X \sim Binomial(n=20, p=0.4)$$, what is $$P(X=8)$$?

Accepted Answer

$$P(X=8) = \binom{20}{8} (0.4)^8$ (0.6$$)^{12}$$

Question 6

A normal random variable $$X$$ has mean $$\mu = 50$$ and standard deviation $$\sigma = 8. $$What is the $$z-$$score for $$X = 66$$?

Accepted Answer

$$z = \frac{66 - 50}{8} = 2$$

Question 7

The 68-95-99.7 rule for the normal distribution states that approximately what percent of observations fall within two standard deviations of the mean?

Accepted Answer

95%

Question 8

A casino models its high-roller losses as $$N(\mu = 850,000, \sigma = 60,000). $$What loss threshold should the casino set to only offer discounts to the top 1% of losers?

Accepted Answer

$$x^* = \mu + z_{0.99} \sigma = 850,000 + 2.326 	imes 60,000 \approx 989,560$$

Question 9

Suppose the fuel efficiency of cars is normally distributed with mean $$\mu = 24$$ mpg and $$\sigma = 6$$ mpg. What percentage of cars get less than 15 mpg?

Accepted Answer

$$P(X \u003c 15) = NORM.DIST(15,24,6,TRUE) \approx 0.0668$$

Question 10

A portfolio consists of 30% stocks ($$\mu_S = 0.115$$, $$\sigma_S = 0.199) $$and 70% bonds ($$\mu_B = 0.044$$, $$\sigma_B = 0.046$$), with returns assumed independent and normal. What is the mean and standard deviation of the portfolio return?

Accepted Answer

$$\mu_P = 0.3 	imes 0.115 + 0.7 	imes 0.044 = 0.0653$$, $$\sigma_P = \sqrt{(0.3^2$ 	imes 0.199^2$) + (0.7^2$ 	imes 0.046^2$)} \approx 0.0678$$

Question 11

If $$X \sim Binomial(n=100, p=0.5)$$, which normal distribution can be used to approximate $$X$$?

Accepted Answer

$$N(\mu = 50, \sigma = 5)$$

Question 12

A normal random variable $$X$$ has $$\mu = 10$$ and $$\sigma = 2. $$What value $$x^*$$ satisfies $$P(X \u003c x^*) = 0.975$$?

Accepted Answer

$$x^* = 10 + 1.96 	imes 2 = 13.92$$

Question 13

Which of the following statements about the binomial distribution is TRUE?

Accepted Answer

The binomial distribution models the number of successes in $$n$$ independent trials, each with the same probability $$p of $$success.

Question 14

If $$X \sim N(0.00351, 0.023)$$, what is $$P(X \u003e 0.03)$$?

Accepted Answer

$$P(X \u003e 0.03) = 1 - NORM.DIST(0.03, 0.00351, 0.023, TRUE) \approx 0.125$$

Question 15

A football player has a probability $$p = 0.661 of $$winning a game. Over a 17-game season, what is the expected number of wins and the standard deviation?

Accepted Answer

$$E(X) = 17 	imes 0.661 = 11.237$$, $$SD(X) = \sqrt{17 	imes 0.661 	imes 0.339} \approx 1.97$$

Question 16

If $$X \sim N(24, 6)$$, what is the probability that a car has fuel efficiency between 20 and 30 mpg?

Accepted Answer

$$P(20 \leq X \u003c 30) = NORM.DIST(30,24,6,TRUE) - NORM.DIST(20,24,6,TRUE) \approx 0.589$$

Question 17

Which of the following is TRUE about the sum of two independent normal random variables $$X_1$$ \sim N(\$$mu_1$$, \$$sigma_1^2$$)$$ and X_2$ \sim N(\mu_2$, \$$sigma_2$$^2)$$?

Accepted Answer

$$X_1$$ + $$X_2$$ \sim N(\$$mu_1$$ + \$$mu_2$$, \$$sigma_1^2$$ + \$$sigma_2^2$$)$$

Binomial and Normal Random Variables: Study Notes for Statistics for Business

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