Question

Consider a random variable \(X\) of the continuous type with cdf \(F(x)\) and pdf \(f(x)\). The hazard rate (or failure rate or force of mortality) is defined by $$ r(x)=\lim _{\Delta \rightarrow 0} \frac{P(x \leq X0\). (a) Show that \(r(x)=f(x) /(1-F(x))\). (b) If \(r(x)=c\), where \(c\) is a positive constant, show that the underlying distribution is exponential. Hence, exponential distributions have constant failure rate over all time. (c) If \(r(x)=c x^{b}\), where \(c\) and \(b\) are positive constants, show that \(X\) has a Weibull distribution, i.e., $$ f(x)=\left\\{\begin{array}{ll} c x^{b} \exp \left\\{-\frac{c x^{b+1}}{b+1}\right\\} & 0

Short answer

(a) The hazard rate \(r(x) = f(x) / (1-F(x))\). (b) When \(r(x) = c\), X follows an exponential distribution. (c) When \(r(x) = c * x^{b}\), X follows a Weibull distribution. (d) When \(r(x) = c * e^{b * x}\), X follows a Gompertz distribution.

Step-by-step solution

Prove the General Equation

Start by noting the definition of conditional probability: \(P(A|B) = P(A \cap B) / P(B)\), and apply it to the definition of hazard rate \(r(x) = P(x ≤ X < x+Δ | X ≥ x) / Δ\), which results in \(r(x) = P(x ≤ X < x+Δ , X ≥ x) / (Δ*P( X ≥ x))\), which simplifies to \(r(x) = P(x ≤ X < x+Δ) / Δ\) as the events \(x ≤ X < x+Δ\) and \(X ≥ x\) are not disjoint. This is equal to \(r(x) = f(x) / (1 - F(x))\) as \(f(x) = F'(x)\) and \(P(X ≥ x) = 1 - F(x)\).

Show the Distribution is Exponential when r(x) = c

When \(r(x) = c\), we have \(f(x) / (1 - F(x)) = c\), which upon rearrangement gives \(F'(x) = c - c*F(x)\). This is a standard first order differential equation with solution \(F(x) = 1 - e^{-cx}\). The derivative of this function gives the pdf \(f(x) = c*e^{-cx}\), which is the exponential distribution.

Show that X has a Weibull Distribution when r(x) = c*x^b

When \(r(x) = c*x^b\), the equation \(f(x) / (1 - F(x)) = c*x^b\) again takes the form of a standard first order differential equation, which leads to the pdf \(f(x) = c*x^{b} * e^{-\frac{c * x^{b+1}}{b+1}}\) for \(x > 0\), which is the Weibull distribution.

Show that X has a Gompertz Distribution when r(x) = c*e^{b*x}

When \(r(x) = c*e^{b*x}\), the equation \(f(x) / (1 - F(x)) = c*e^{b*x}\) leads to the cdf \(F(x) = 1 - e^{\frac{c}{b}*(1 - e^{b*x})}\) for \(x > 0\), which is the Gompertz distribution.

Key concepts

Exponential Distribution

The exponential distribution is a continuous probability distribution commonly used to model the time until an event occurs—such as the failure of a machine or the time between events in a Poisson process. The key feature of an exponential distribution is its constant hazard rate. This means that the probability of the event occurring in the next instant is the same regardless of how much time has already passed.

Mathematically, if the hazard rate is constant given by a positive constant \( c \), the exponential distribution can be defined. Using the relation \( r(x) = \frac{f(x)}{1 - F(x)} = c \), where \( f(x) \) is the probability density function (pdf) and \( F(x) \) is the cumulative distribution function (cdf), it simplifies to show that:

• \( F(x) = 1 - e^{-cx} \), which indicates the cdf.
• \( f(x) = c e^{-cx} \), which depicts the pdf.

Both these functions show how the exponential distribution has a memoryless property, ensuring that the probability remains consistent over time.

Weibull Distribution

The Weibull distribution generalizes the exponential distribution by introducing another parameter, \( b \), that allows for a varying hazard rate. It is typically used in reliability analysis and survival studies because it can model different forms of failure rates, including increasing, decreasing, or constant.

When the hazard rate is given by \( r(x) = c x^b \), where \( c \) and \( b \) are positive constants, it is indicative of a Weibull distribution. Applying this hazard rate into the definition \( \frac{f(x)}{1-F(x)} = c x^b \), we derive the probability density function:

\[ f(x) = c x^b e^{-\frac{c x^{b+1}}{b+1}} \]

For different values of \( b \), the shape of the distribution can vary:

• When \( b = 1 \), it simplifies to the exponential distribution.
• \( b < 1 \) indicates a decreasing failure rate.
• \( b > 1 \) suggests an increasing failure rate.

This flexibility makes the Weibull distribution highly adaptable to different types of life data.

Gompertz Distribution

The Gompertz distribution is widely used in the context of modeling "length of life." It accounts for scenarios where the hazard rate exponentially increases over time, which corresponds to many biological organisms whose mortality rate accelerates as they age.

In the case of this distribution, the hazard rate is expressed as \( r(x) = c e^{b x} \), with \( c \) and \( b \) being positive constants. This results in a cumulative distribution function (cdf) of the form:

\[ F(x) = 1 - e^{\frac{c}{b}(1 - e^{b x})} \]

This particular structure showcases an exponentially increasing hazard rate, making the Gompertz distribution ideal for actuarial science and demography for life expectancy projections. The faster rise in the hazard rate over time reflects a greater likelihood of the event (i.e., survival time) as age progresses.

In practical applications, it conforms well to datasets involving human lifespan, aging processes, and even financial risk modeling.