Problem 20 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

The interest rates paid by 30 financial institutions on a certain day for money market deposit accounts are shown in the accompanying table: $$\begin{array}{lcccc}\hline \text { Rate, \% } & 6 & 6.25 & 6.55 & 6.56 \\ \hline \text { Institutions } & 1 & 7 & 7 & 1 \\ \hline\end{array}$$ $$\begin{array}{lcccc} \hline \text { Rate, } \% & 6.58 & 6.60 & 6.65 & 6.85 \\\\\hline \text { Institutions } & 1 & 8 & 3 & 2 \\ \hline\end{array}$$ Let the random variable \(X\) denote the interest rate paid by a randomly chosen financial institution on its money market deposit accounts and find the probability distribution associated with these data.

Step-by-Step Solution

Answer

The probability distribution for the interest rates is as follows: $$ \begin{array}{c|c} \text{Interest Rate (\%)} & \text{Probability} \\ \hline 6.00 & \frac{1}{30} \\ 6.25 & \frac{7}{30} \\ 6.55 & \frac{7}{30} \\ 6.56 & \frac{1}{30} \\ 6.58 & \frac{1}{30} \\ 6.60 & \frac{8}{30} \\ 6.65 & \frac{3}{30} \\ 6.85 & \frac{2}{30} \\ \end{array} $$

Step 1: Identify the total number of financial institutions

To calculate the total number of financial institutions, we sum the number of institutions for each interest rate as given in the table: Total Institutions = 1 + 7 + 7 + 1 + 1 + 8 + 3 + 2 = 30

Step 2: Calculate probabilities for each interest rate

Using the total number of financial institutions found in Step 1, calculate the probability of each interest rate. The probability of an interest rate is given by the number of institutions offering that interest rate divided by the total number of institutions. Probability for 6% = 1 / 30 Probability for 6.25% = 7 / 30 Probability for 6.55% = 7 / 30 Probability for 6.56% = 1 / 30 Probability for 6.58% = 1 / 30 Probability for 6.60% = 8 / 30 Probability for 6.65% = 3 / 30 Probability for 6.85% = 2 / 30

Step 3: Present the probability distribution

Using the probabilities calculated in step 2 and the interest rates, we can now present the probability distribution associated with these data: $$ \begin{array}{c|c} \text{Interest Rate (\%)} & \text{Probability} \\ \hline 6.00 & \frac{1}{30} \\ 6.25 & \frac{7}{30} \\ 6.55 & \frac{7}{30} \\ 6.56 & \frac{1}{30} \\ 6.58 & \frac{1}{30} \\ 6.60 & \frac{8}{30} \\ 6.65 & \frac{3}{30} \\ 6.85 & \frac{2}{30} \\ \end{array} $$

Key Concepts

Random Variables

In probability and statistics, a random variable is a variable that represents a numerical outcome of a random process. It's essentially how we quantify the result of some random phenomenon. When we say random, we mean the result can vary based on some inherent randomness in the process.
In this exercise, our random variable, denoted by \(X\), is the interest rate offered by financial institutions on a specific day. Each interest rate is an outcome that \(X\) can take, and the number of institutions offering each rate helps us determine how likely it is for \(X\) to take that particular value.
This variability is precisely captured by finding the probability distribution for \(X\), allowing us to see not only the possible rates but also their corresponding likelihoods of occurrence. Notably, random variables can take on discrete values, like specific interest rates, or continuous values in some instances.

Interest Rates

Interest rates are pivotal in finance, acting as the cost of borrowing money or the reward for saving it. They're often expressed as a percentage of the principal, which is the initial amount of money either borrowed or saved.
The exercise illustrates different interest rates provided by financial institutions for money market deposit accounts. These accounts offer liquidity and a higher interest compared to regular savings accounts.
Different financial institutions might offer varying rates due to numerous factors:

• Risk assessment of the institution.
• Market conditions and demand.
• Regulatory environment and policies.

Understanding how interest rates vary and the probability of a certain rate being offered can help consumers and investors make informed decisions about where to save or invest their money.

Financial Institutions

Financial institutions play a crucial role in the economy by acting as intermediaries between savers and borrowers. They include banks, credit unions, insurance companies, and investment firms.
In this exercise, we observe how different financial institutions offer varying interest rates on money market deposit accounts. These institutions determine interest rates based on their financial strategies, policies, and competitiveness in attracting customers.
They assess:

• The need to balance the attraction of new deposits while managing the costs of lending.
• Economic conditions, such as inflation and policy changes.
• Liquidity requirements to ensure they can meet customer demands.

Understanding the interest rates offered by financial institutions can give us insight into their market strategies and economic outlook. These rates can influence consumers’ decisions on where to deposit their money for optimal returns.