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2、The lower limit of a normal distribution is:

A) negative infinity.

B) zero.

C) one.

D) negative one.

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The correct answer is A

By definition, a true normal distribution has a positive probability density function from negative to positive infinity.

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3、Which of the following statements about a normal distribution is least accurate?

A) A normal distribution has excess kurtosis of three.

B) The mean, median, and mode are equal.

C) The mean and variance completely define a normal distribution.

D) Approximately 68% of the observations lie within +/- 1 standard deviation of the mean.

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The correct answer is B

Let S denote success, U denote unsuccessful, and F denote a favorable report. We have P(S)=0.30, P(U)=0.70, P(F | S)=0.70, and P(F | U)=0.10. Using Bayes theorem, the probability of a successful contract, given a favorable report is:[attach]13843[/attach]

 

 

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AIM 5: Define probability mass function, probability function, probability density function, cumulative distribution function.

Which of the following statements about probability distributions is least accurate?

A) In a binomial distribution each observation has only two possible outcomes that are mutually exclusive.

B) A probability distribution is, by definition, normally distributed.

C) A probability distribution includes a listing of all the possible outcomes of an experiment.

D) One of the key properties of a probability function is  0 ≤ p ≤ 1.

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The correct answer is B

Probabilities must be zero or positive, but a probability distribution is not necessarily normally distributed. Binomial distributions are either successes or failures.

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3、A major securities exchange is considering the introduction of a new derivative contract. In the past, the success rate for new derivatives has been 30 percent. Extensive market research has produced a positive marketing research report for the contract under consideration. Historically, 70 percent of the successful contracts have received favorable reports prior to introduction. Only 10 percent of unsuccessful contracts have received favorable reports. What is the probability that the new contract will be successful?

A) 87%.

B) 75%.

C) 13%.

D) 25%.

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AIM 4: Explain Bayes’ theorem and use Bayes’ formula to determine the probability of causes for a given event.

1、Bonds rated B have a 25% chance of default in five years. Bonds rated CCC have a 40% chance of default in five years. A portfolio consists of 30% B and 70% CCC-rated bonds. If a randomly selected bond defaults in a five-year period, what is the probability that it was a B-rated bond?

A) 0.625.

B) 0.429.

C) 0.211.

D) 0.250.

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The correct answer is C

According to Bayes' formula: P(B/default) = P(default and B)/P(default).

P(default and B )= P(default/B) × P(B) = 0.250 × 0.300 = 0.075

P(default and CCC) = P(default/CCC) × P(CCC) = 0.400 × 0.700 = 0.280

P(default) = P(default and B) + P(default and CCC) = 0.355

P(B/default) = P(default and B)/P(default) = 0.075 / 0.355 = 0.211

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2、The probability of A is 0.4. The probability of AC is 0.6. The probability of (B|A) is 0.5, and the probability of (B|AC) is 0.2. Using Bayes’ formula, what is the probability of (A|B)?

A) 0.875.

B) 0.125.

C) 0.375.

D) 0.625.