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Which of the following is least likely to be an example of a discrete random variable?

A) The rate of return on a real estate investment.

B) The number of days of sunshine in the month of May 2006 in a particular city.

C) Quoted stock prices on the NASDAQ.

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The rate of return on a real estate investment, or any other investment, is an example of a continuous random variable because the possible outcomes of rates of return are infinite (e.g., 10.0%, 10.01%, 10.001%, etc.). Both of the other choices are measurable (countable).

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A random variable that has a countable number of possible values is called a:

A) probability distribution.

B) continuous random variable.

C) discrete random variable.

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A discrete random variable is one for which the number of possible outcomes are countable, and for each possible outcome, there is a measurable and positive probability. A continuous random variable is one for which the number of outcomes is not countable.

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A probability function:

A) is often referred to as the "cdf."

B) specifies the probability that the random variable takes on a specific value.

C) only applies to continuous distributions.

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This is true by definition.

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Which of the following could least likely be a probability function?

A) X1,2,3,4) p(x) = 0.2.

B) X1,2,3,4) p(x) = (x × x) / 30.

C) X1,2,3,4) p(x) = x / 10.

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In a probability function, the sum of the probabilities for all of the outcomes must equal one. Only one of the probability functions in these answers fails to sum to one.

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If a smooth curve is to represent a probability density function, what two requirements must be satisfied? The area under the curve must be:

A) one and the curve must not fall below the horizontal axis.

B) one and the curve must not rise above the horizontal axis.

C) zero and the curve must not fall below the horizontal axis.

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If a smooth curve is to represent a probability density function, the total area under the curve must be one (probability of all outcomes equals 1) and the curve must not fall below the horizontal axis (no outcome can have a negative chance of occurring).

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In a continuous probability density function, the probability that any single value of a random variable occurs is equal to what?

A) One.

B) Zero.

C) 1/N.

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Since there are infinite potential outcomes in a continuous pdf, the probability of any single value of a random variable occurring is 1/infinity = 0.

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A cumulative distribution function for a random variable X is given as follows:

x	F(x)
5	0.14
10	0.25
15	0.86
20	1.00

The probability of an outcome less than or equal to 10 is:

A) 14%.

B) 25%.

C) 39%.

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A cumulative distribution function (cdf) gives the probability of an outcome for a random variable less than or equal to a specific value. For the random variable X, the cdf for the outcome 10 is 0.25, which means there is a 25% probability that X will take a value less than or equal to 10.

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Which of the following qualifies as a cumulative distribution function?

A) F(1) = 0, F(2) = 0.25, F(3) = 0.50, F(4) = 1.

B) F(1) = 0, F(2) = 0.5, F(3) = 0.5, F(4) = 0.

C) F(1) = 0.5, F(2) = 0.25, F(3) = 0.25.

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Because a cumulative probability function defines the probability that a random variable takes a value equal to or less than a given number, for successively larger numbers, the cumulative probability values must stay the same or increase.

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A random variable X is continuous and bounded between zero and five, X0 ≤ X ≤ 5). The cumulative distribution function (cdf) for X is F(x) = x / 5. Calculate P(2 ≤ X ≤ 4).

A) 1.00.

B) 0.40.

C) 0.50.

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For a continuous distribution, P(a ≤ X ≤b) = F(b) − F(a). Here, F(4) = 0.8 and F(2) = 0.4. Note also that this is a uniform distribution over 0 ≤ x ≤ 5 so Prob(2 < x < 4) = (4 − 2) / 5 = 40%.

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Which of the following random variables would be most likely to follow a discrete uniform distribution?

A) The outcome of a roll of a standard, six-sided die where X equals the number facing up on the die.

B) The number of heads on the flip of two coins.

C) The outcome of the roll of two standard, six-sided dice where X is the sum of the numbers facing up.

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The discrete uniform distribution is characterized by an equal probability for each outcome. A single die roll is an often-used example of a uniform distribution. In combining two random variables, such as coin flip or die roll outcomes, the sum will not be uniformly distributed.