Probability Distribution of Random Variables

Learning objectives: distinguish between discrete and continuous random variable; compute statistics about random variable; compute statistics about a function of a random variable; compute statistics about the sum of a linear composite of random variables; identify which type of distribution a given random variable is most likely to follow; solve problems involving standard distributions manually using formulas; solve business problems involving standard distributions using spreadsheet templates.

3-1
The number of the phone calls arriving at an exchange during any given minute between noon and 1:00 pm on a weekday is a random variable with the following probability distribution.
X P(x)
0 0.3
1 0.2
2 0.2
3 0.1
4 0.1
5 0.1
a) Verify that P(x) is a probability distribution
b) Find the cumulative distribution function of the random variable
c) Use the cumulative distribution function to find the probability that between 12:34 and 12:35 PM more than two calls will arrive at the exchange.
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3-6
The number of intercity shipment orders arriving daily at a transportation company is a random variable X with the following probability distribution:
X P(x)
0 0.1
1 0.2
2 0.4
3 0.1
4 0.1
5 0.1
a) Verify that P(x) is a probability distribution
b) Find the cumulative probability function of X
c) Use the cumulative probability function computed in (b) to find the probability that anywhere from one to four shipment orders will arrive on a given day.
d) When more than three orders arrive on a given day, the company incurs additional costs due to the need to hire extra drivers and loaders. What is the probability that extra costs will be incurred on a given day?
e) Assuming that the number of orders arriving on different days is independent of each other, what is the probability that no orders will be received over a period of five working days?
f) Again assuming independence of orders on different days, what is the probability that extra cost will be incurred two days in a row?
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3-9
Returns on investments overseas, especially in Europe and the Pacific Rim, are expected to be higher than those of US markets in the near term, and analysts are now recommending investments in international portfolios. An investment consultant believes that the probability distribution of returns (in percent per year) on one such portfolio is as follows:
X (%) P(x)
9 0.05
10 0.15
11 0.30
12 0.20
13 0.15
14 0.10
15 0.05
a) Verify that P(x) is a probability distribution
b) What is the probability that returns will be at least 12%?
c) Find the cumulative distribution of returns.
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3-17
Find the mean, variance and standard deviation of the annual income of a hedge fund manager, using the probability distribution in problem 3-7.
X( 5 millions) P(x)
$1,700 0.2
1,500 0.2
1,200 0.3
1,000 0.1
800 0.1
600 0.05
400 0.05
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3-39
An MBA graduate is applying for a nine jobs, and believes that she has in each of the nine cases a constant and independent 0.48 probability of getting an offer.
a) What is the probability that she will have at least three offers?
b) If she wants to be 95% confident of having at least three offers, how many more jobs should she apply for? (Assume each of these additional applications will also have the same probability of success.)
c) If there are no more than the original nine jobs that she can apply for, what value of probability of success would give her 95% confidence of at least three offers?
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