The Troubled Asset Relief Program (TARP), passed by the U.S....

The Troubled Asset Relief Program (TARP), passed by the U.S. Congress in October 2008 . provided $\$ 700$ billion in assistance for the struggling U.S. economy. Over $\$ 200$ billion was given to troubled financial institutions with the hope that there would be an increase in lending to help jump-start the economy. But three months later, a Federal Reserve survey found that two-thirds of the banks that had received TARP funds had tightened terms for business loans (The Wall Street Journal, February $3,2009$ ). Of the 10 banks that were the biggest recipients of TARP funds, only 3 had actually increased lending this period. For the purposes of this exercise, assume that you will randomly select 3 of these 10 banks for a study that will continue to monitor bank lending practices. Let $x$ be a random variable indicating the number of banks in the study that had increased lending. a. What is $f(0) ?$ What is your interpretation of this value? b. What is $f(3) ?$ What is your interpretation of this value? c. Compute $f(1)$ and $f(2) .$ Show the probability distribution for the number of banks in the study that had increased lending. What value of $x$ has the highest probability? d. What is the probability that the study will have at least one bank that had increased lending? e. Compute the expected value, variance, and standard deviation for the random variable.

The Troubled Asset Relief Program (TARP), passed by the U.S. Congress in October 2008 . provided $\$ 700$ billion in assistance for the struggling U.S. economy. Over $\$ 200$ billion was given to troubled financial institutions with the hope that there would be an increase in lending to help jump-start the economy. But three months later, a Federal Reserve survey found that two-thirds of the banks that had received TARP funds had tightened terms for business loans (The Wall Street Journal, February $3,2009$ ). Of the 10 banks that were the biggest recipients of TARP funds, only 3 had actually increased lending this period.

For the purposes of this exercise, assume that you will randomly select 3 of these 10 banks for a study that will continue to monitor bank lending practices. Let $x$ be a random variable indicating the number of banks in the study that had increased lending.
a. What is $f(0) ?$ What is your interpretation of this value?
b. What is $f(3) ?$ What is your interpretation of this value?
c. Compute $f(1)$ and $f(2) .$ Show the probability distribution for the number of banks in the study that had increased lending. What value of $x$ has the highest probability?
d. What is the probability that the study will have at least one bank that had increased
lending?
e. Compute the expected value, variance, and standard deviation for the random variable.

Key Concepts

Complement Rule

The complement rule in probability is the principle that the probability of an event occurring is equal to one minus the probability of it not occurring. This rule is particularly useful when calculating the probability of 'at least one' success by subtracting the probability of 'zero' successes from one.

Variance and Standard Deviation

These are measures of dispersion in a probability distribution. Variance indicates the average squared deviation from the expected value, while the standard deviation is the square root of the variance, offering insight into the spread or variability of the outcomes in the same unit as the original variable.

Expected Value

The expected value of a random variable is the long-run average or mean value it would take over many trials. In the setting of a hypergeometric distribution, it is computed by weighting each possible outcome by its probability, and it provides a measure of the central tendency of the distribution.

Combinatorial Analysis

Combinatorial analysis involves counting the number of ways events can occur, typically using combinations. When dealing with probabilities in sampling scenarios, combinations are used to compute both the total number of possible samples and the number of favorable outcomes, which is essential in forming the probability mass function for the hypergeometric distribution.

Hypergeometric Distribution

This is a probability distribution used for situations where a fixed number of items are drawn from a finite population without replacement, and where the population contains a known number of successes and failures. The distribution calculates the probability of drawing a specific number of successes using combinations, making it ideal for problems that involve sampling without replacement.

Probability Mass Function (PMF)

The PMF is a function that gives the probability that a discrete random variable is exactly equal to some value. In the context of the hypergeometric distribution, it is calculated using the formula that involves combinations, reflecting the probability of selecting a particular number of successes from the sample.

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