Q: Suppose a lottery ticket costs $$1.00 and the probability of winning 1$$,000,000 is $$0.000001. $$What is the expected gain from buying one ticket?

$$E(X) = 1,000,000 \times 0.000001 + (-1) \times 0.999999 = 1 - 0.999999 = 0.000001$$

Q: Suppose the mean SAT Evidence-Based Reading and Writing score is $$520$$ with a standard deviation of $$110. $$What is the SAT score corresponding to a z-score of $$-2$$?

$$x = 520 - 2 \times 110 = 300$$

Question 1

Which of the following best describes a discrete random variable?

Accepted Answer

A variable that can assume only a finite or countable number of values.

Question 2

Given Benford’s law for the first digit X of financial records, with probabilities $$p(1) = 0.301$$, $$p(2) = 0.176$$, $$p(3) = 0.125$$, $$p(4) = 0.097$$, $$p(5) = 0.079$$, $$p(6) = 0.067$$, $$p(7) = 0.058$$, $$p(8) = 0.051$$, $$p(9) = 0.046$$, what is the probability that the first digit is at least 5?

Accepted Answer

$$0.079 + 0.067 + 0.058 + 0.051 + 0.046 = 0.301$$

Question 3

Which of the following is a property of the probability distribution of a discrete random variable X?

Accepted Answer

The sum of all probabilities $$p(x)$$ equals 1.

Question 4

Suppose a lottery ticket costs $$1.00 and the probability of winning 1$$,000,000 is $$0.000001. $$What is the expected gain from buying one ticket?

Accepted Answer

$$E(X) = 1,000,000 	imes 0.000001 + (-1) 	imes 0.999999 = 1 - 0.999999 = 0.000001$$

Question 5

For a continuous random variable, what does the area under the probability density curve between two values represent?

Accepted Answer

The probability that the variable falls within that interval.

Question 6

A density curve must satisfy which of the following properties?

Accepted Answer

It must have area exactly 1 underneath it.

Question 7

According to the Empirical Rule (68-95-99.7 Rule), what percentage of observations in a normal distribution fall within two standard deviations of the mean?

Accepted Answer

95%

Question 8

If the mean and median of a density curve are not equal, what can you infer about the distribution?

Accepted Answer

The distribution is skewed.

Question 9

Given a normal distribution $$N(66, 8)$$ for battery life (in months), what is the z-score for a battery that lasts 58 months?

Accepted Answer

$$z = \frac{58 - 66}{8} = -1$$

Question 10

Suppose the mean SAT Evidence-Based Reading and Writing score is $$520$$ with a standard deviation of $$110. $$What is the SAT score corresponding to a z-score of $$-2$$?

Accepted Answer

$$x = 520 - 2 	imes 110 = 300$$

Question 11

Which of the following statements about the standard normal distribution is TRUE?

Accepted Answer

It has mean 0 and standard deviation 1.

Question 12

A normal probability plot is used to:

Accepted Answer

Determine if data plausibly come from a normal distribution.

Question 13

Refer to the first image (three density curves with balance points). Which curve represents a symmetric distribution?

Accepted Answer

The middle curve.

Question 14

Refer to the second image (probability plot for Carbonate). What does the straight-line pattern in the plot indicate?

Accepted Answer

The data follow a normal distribution.

Question 15

Refer to the third image (probability plot for Siliciclastic Bedrock). What does the curved and segmented pattern in the plot suggest?

Accepted Answer

The data are likely from several distinct populations and not normal.

Question 16

Suppose the lifetime of a light bulb is modeled by a continuous uniform distribution from 0 to 1000 hours. What is the probability that a randomly selected bulb lasts between 200 and 400 hours?

Accepted Answer

$$\frac{400 - 200}{1000 - 0} = 0.2$$

Question 17

If $$X \sim N(100, 15)$$, what is the minimum value of $$X$$ such that only 10% of the population scores higher than $$X$$?

Accepted Answer

$$X = 100 + 1.28 	imes 15 = 119.2$$

Question 18

Which of the following is NOT a characteristic of a probability distribution for a continuous random variable?

Accepted Answer

The sum of the probabilities for all possible values equals 1.

Probability Distributions and the Normal Distribution

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