5. An investment firm offers its customers municipal bonds that mature after varying numbers of years. Given that the cumulative distribution function of T, the number of years to maturity for a randomly selected bond, is given by... please solve all parts to this problem. ### Municipal Bonds and Their Maturity: Understanding Probability with Cumulative Distribution FunctionsAn investment firm offers its customers municipal bonds that mature after varying numbers of years. Given that the cumulative distribution function (CDF) of $ T $, the number of years to maturity for a randomly selected bond, is given by $ F(t) $, calculate the following probabilities:- (a) $ P(T = 3) $- (b) $ P(T > 6) $- (c) $ P(1.2 < T < 5) $- (d) $ P(T \leq 3 \mid T \geq 2) $The cumulative distribution function $ F(t) $ is defined as follows:\[F(t) = \begin{cases} 0, & t < 1, \\\frac{1}{4}, & 1 \leq t < 3, \\\frac{1}{2}, & 3 \leq t < 6, \\\frac{3}{4}, & 6 \leq t < 8, \\1, & t \geq 8 \end{cases}\]### Explanation of the CDFThis CDF describes the cumulative probability that the bond will mature in $ t $ or fewer years. - **For $ t < 1 $:** The probability is 0, meaning no bonds mature before 1 year.- **For $ 1 \leq t < 3 $:** The probability is $ \frac{1}{4} $, indicating a 25% chance that the bond matures between 1 and 3 years.- **For $ 3 \leq t < 6 $:** The probability is $ \frac{1}{2} $, indicating a 50% chance that the bond matures by 6 years.- **For $ 6 \leq t < 8 $:** The probability increases to $ \frac{3}{4} $, or 75% chance.- **For $ t \geq 8 $:** The probability becomes 1, indicating all bonds mature by 8 years or later.To find the probabilities requested, utilize the properties and values of the CDF within the specified ranges.

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An investment firm offers its customers municipal bonds that mature after varying numbers of years. Given that the cumulative distribution function of T, the number of years to maturity for a randomly selected bond, is given by F(t), find (a) P(T = 3); (b) P(T>6); (c) P(1.2

An investment firm offers its customers municipal bonds that mature after varying numbers of years. Given that the cumulative distribution function of T, the number of years to maturity for a randomly selected bond, is given by F(t), find (a) P(T = 3); (b) P(T>6); (c) P(1.2

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

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Problem 1P

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5. An investment firm offers its customers municipal bonds that mature after varying numbers of years. Given that the cumulative distribution function of T, the number of years to maturity for a randomly selected bond, is given by... please solve all parts to this problem.

### Municipal Bonds and Their Maturity: Understanding Probability with Cumulative Distribution Functions

An investment firm offers its customers municipal bonds that mature after varying numbers of years. Given that the cumulative distribution function (CDF) of $ T $, the number of years to maturity for a randomly selected bond, is given by $ F(t) $, calculate the following probabilities:

- (a) $ P(T = 3) $
- (b) $ P(T > 6) $
- (c) $ P(1.2 < T < 5) $
- (d) $ P(T \leq 3 \mid T \geq 2) $

The cumulative distribution function $ F(t) $ is defined as follows:

\[
F(t) =
\begin{cases}
0, & t < 1, \\
\frac{1}{4}, & 1 \leq t < 3, \\
\frac{1}{2}, & 3 \leq t < 6, \\
\frac{3}{4}, & 6 \leq t < 8, \\
1, & t \geq 8
\end{cases}
\]

### Explanation of the CDF

This CDF describes the cumulative probability that the bond will mature in $ t $ or fewer years.

- **For $ t < 1 $:** The probability is 0, meaning no bonds mature before 1 year.
- **For $ 1 \leq t < 3 $:** The probability is $ \frac{1}{4} $, indicating a 25% chance that the bond matures between 1 and 3 years.
- **For $ 3 \leq t < 6 $:** The probability is $ \frac{1}{2} $, indicating a 50% chance that the bond matures by 6 years.
- **For $ 6 \leq t < 8 $:** The probability increases to $ \frac{3}{4} $, or 75% chance.
- **For $ t \geq 8 $:** The probability becomes 1, indicating all bonds mature by 8 years or later.

To find the probabilities requested, utilize the properties and values of the CDF within the specified ranges.

Expression, rule, or law that gives the relationship between an independent variable and dependent variable. Some important types of functions are injective function, surjective function, polynomial function, and inverse function.

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