91Ó°ÊÓ

The Cumulative Distribution Function (CDF) is a tool used to describe the probability of a random variable taking a value less than or equal to a specified value. In this exercise, we are dealing with a Weibull distributed variable. The CDF for the Weibull distribution is represented by \( F_Y(y) = 1 - e^{-(y/\beta)^\alpha} \) for values \( y \geq 0 \).

Given our variable \( X \) with a minimum return time of \( \gamma = 3.5 \), the additional time \( Y = X - 3.5 \) follows a Weibull distribution. Therefore, we adjust the CDF for our specific context to find \( F_X(x) \). We plug \( x - 3.5 \) into the Weibull CDF formula:

• For \( x \geq 3.5 \), \( F_X(x) = 1 - e^{-((x-3.5)/1.5)^2} \)
• For \( x < 3.5 \), \( F_X(x) = 0 \) since the return cannot occur before the minimum time.

This CDF helps us understand the behavior of the return times and calculate probabilities for specific time intervals.

The Expected Value or mean of a random variable gives us the average outcome we might expect. For a Weibull distribution with parameters \( \beta \) and \( \alpha \), the expected value \( E(Y) \) is calculated as \( \beta \Gamma(1 + 1/\alpha) \).

In this exercise, \( \beta = 1.5 \) and \( \alpha = 2 \). The mean is then found as

• \( E(Y) = 1.5 \times \Gamma(1.5) \approx 1.32934 \)

Adding the minimum time \( 3.5 \) weeks gives \( E(X) = E(Y) + 3.5 \approx 4.82934 \).

The variance provides a measure of the spread of the distribution. For the Weibull distribution, it's determined by \( V(Y) = \beta^2 (\Gamma(1 + 2/\alpha) - (\Gamma(1 + 1/\alpha))^2) \). For our parameters, this simplifies to

• \( V(Y) \approx 0.520725 \)

Since \( X - 3.5 = Y \), the variance of \( X \) remains \( V(X) = V(Y) \approx 0.520725 \). The expected value and variance are crucial for assessing the general return patterns and variability.

In probability, we may often want to find how likely certain outcomes are. For variable \( X \), we are interested in probabilities like \( P(X > 5) \) or \( P(5 \leq X \leq 8) \). These calculations employ the CDF we derived earlier.

To compute \( P(X > 5) \), we need \( 1 - F_X(5) \). Inside the formula, plug \( (5 - 3.5)/1.5 \) into the Weibull CDF:

• \[ F_X(5) = 1 - e^{-0.4444} \approx 0.36093 \]
• Hence, \( P(X > 5) \approx 0.63907 \)

For \( P(5 \leq X \leq 8) \), find \( F_X(8) - F_X(5) \):

• Compute \( F_X(8) \approx 0.99996 \) by plugging into the CDF
• \[ P(5 \leq X \leq 8) \approx 0.99996 - 0.36093 = 0.63903 \]

These probability calculations help assess return odds within specific time ranges and are valuable for decision-making processes.