91Ó°ÊÓ

The hazard rate function, often referred to as the failure rate in reliability theory, represents the instantaneous rate of occurrence of an event at time \( t \), given that the event has not occurred before \( t \). It is particularly useful in fields like survival analysis and reliability engineering. For a continuous random variable \( Z \), with cumulative distribution function (CDF) \( F_Z(z) \) and probability density function (PDF) \( f_Z(z) \), the hazard rate function \( \lambda_Z(t) \) is defined as:

• \( \lambda_Z(t) = \frac{f_Z(t)}{1 - F_Z(t)} \)

Let's break down this formula:

• \( f_Z(t) \) is the PDF, showing the likelihood of \( Z \) taking a value close to \( t \).
• \( 1 - F_Z(t) \) represents the probability that \( Z \) is greater than \( t \), indicating the "survival" past time \( t \).

In simple words, the hazard rate is the chance per unit time that the event will happen right after \( t \), given it didn't happen before. This function helps differentiate between different types of distributions based on how likely an event is to occur at a given moment.

The cumulative distribution function (CDF) of a random variable gives us the probability that the variable will take a value less than or equal to a certain number. For a random variable \( X \), the CDF is noted as \( F_X(x) \), and it reflects the cumulative probability up to the point \( x \). The formula is:

• \( F_X(x) = P(X \leq x) \)

The CDF is crucial because it encompasses all the probabilities of outcomes, effectively summarizing the distribution of a random variable. Moreover, the CDF is a non-decreasing function, starting at 0 and eventually reaching 1 as \( x \) approaches infinity.

When working with two independent random variables, such as in our exercise where \( W = \min(X, Y) \), the CDF of \( W \) is derived using the relation between \( X \) and \( Y \). Specifically, for such a scenario, the CDF of \( W \), \( F_W(w) \), is computed as:

• \( F_W(w) = 1 - (1 - F_X(w))(1 - F_Y(w)) \)

This formula arises by considering the complementary probabilities of both \( X \) and \( Y \) being greater than \( w \). It's a clear example of how the concept of minimum translates into probability terms for cumulative events.

Two random variables are independent when the occurrence of one does not affect the probability of occurrence of another. In probabilistic terms, this means that knowing the outcome of one random variable gives no information about the other. Mathematically, two random variables \( X \) and \( Y \) are independent if:

• \( P(X \leq x, Y \leq y) = P(X \leq x) \cdot P(Y \leq y) \)

This property of independence simplifies many calculations, such as determining joint probabilities, as shown in the exercise:

• \( P(X > w, Y > w) = P(X > w) \cdot P(Y > w) \)

In the context of the exercise, this independence is crucial in the calculation of the CDF of \( W = \min(X, Y) \) and in determining the hazard rate function. It allows us to separately consider each variable's contribution to the overall behavior of \( W \). The independence leads directly to straightforward computation for functions like minimum or maximum, where each event's probability factors without interference from the others.