An investment firm offers its customers municipal bonds that...

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 $$ F(t)=\left\{\begin{array}{ll} 0, & t<1 \\ \frac{1}{4}, & 1 \leq t<3 \\ \frac{1}{2}, & 3 \leq t<5 \\ \frac{3}{4}, & 5 \leq t<7 \\ 1, & t \geq 7 \end{array}\right. $$ find (a) $P(T=5)$ (b) $P(T>3)$ (c) $P(1.4<T<6)$ (d) $P(T \leq 5 \mid T \geq 2)$

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
$$
F(t)=\left\{\begin{array}{ll}
0, & t<1 \\
\frac{1}{4}, & 1 \leq t<3 \\
\frac{1}{2}, & 3 \leq t<5 \\
\frac{3}{4}, & 5 \leq t<7 \\
1, & t \geq 7
\end{array}\right.
$$
find
(a) $P(T=5)$
(b) $P(T>3)$
(c) $P(1.4<T<6)$
(d) $P(T \leq 5 \mid T \geq 2)$

Key Concepts

Probability Calculation Using CDF Differences

For discrete random variables, the probability that the variable equals a specific value is determined by the difference between the CDF values immediately after and immediately before that value. Similarly, probabilities over intervals can be obtained by subtracting the value of the CDF at the start of the interval from its value at the end of the interval. This technique is essential for extracting precise probability values from a given CDF.

Conditional Probability

Conditional probability measures the likelihood of an event occurring given that another event has already occurred. It is calculated by taking the joint probability of both events and dividing it by the probability of the conditioning event. In problems involving CDFs, this often involves computing the necessary probabilities using the CDF and then applying the conditional probability formula to find the desired result.

Cumulative Distribution Function (CDF)

The CDF of a random variable gives the probability that the variable will take a value less than or equal to a specific value. In problems like this, the CDF is defined piecewise, providing cumulative probabilities over different intervals. Understanding the CDF is crucial because it encapsulates the overall behavior of the probability distribution and is used to calculate probabilities for both ranges and specific points.

Discrete Random Variable and Jump Discontinuity

When the CDF is piecewise constant with jumps at certain points, it indicates that the random variable is discrete and has specific values with nonzero probability. Each jump in the CDF corresponds to a probability mass located at that particular point. Recognizing these jumps allows one to determine the probability of the random variable taking on a specific value.

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