Problem 68 - Calculus for Biology and Medicine

Question

The hazard-rate function of an organism is given by $$ \lambda(x)=0.04 x^{3.1}, \quad x \geq 0 $$ where \(x\) is measured in years. (a) What is the probability that the organism will live for more than three years? (b) What is the probability that the organism will live for another three years, given that it survived the first three years?

Step-by-Step Solution

Answer

(a) 0.4513, (b) 0.0095

Step 1: Identify the survival function

To find the probability that the organism will live for more than three years, we first need to determine the survival function, \( S(x) \). The relationship between the hazard rate function and the survival function is given by: \[ S(x) = e^{-\int_0^x \lambda(t) \, dt} \] where \( \lambda(t) \) is the hazard-rate function.

Step 2: Integrate the hazard-rate function

Integrate the given hazard-rate function \( \lambda(x) = 0.04 x^{3.1} \) from 0 to \( x \):\[ \int_0^x 0.04 t^{3.1} \, dt = 0.04 \left[ \frac{t^{4.1}}{4.1} \right]_0^x = 0.04 \cdot \frac{x^{4.1}}{4.1} \] Simplifying gives:\[ 0.04 \cdot \frac{x^{4.1}}{4.1} = \frac{0.04}{4.1} x^{4.1} \]

Step 3: Substitute into the survival function

Now, substitute the result into the survival function formula:\[ S(x) = e^{-\frac{0.04}{4.1} x^{4.1}} \] This is the expression for the survival function for the organism.

Step 4: Calculate probability for more than three years

To find the probability that the organism will live for more than three years, plug \( x = 3 \) into the survival function:\[ S(3) = e^{-\frac{0.04}{4.1} \times 3^{4.1}} \] Calculate the value:\[ S(3) \approx e^{-\frac{0.04}{4.1} \times 81.5731} \] Use a calculator to find the numerical value:\[ S(3) \approx e^{-0.7951} \approx 0.4513 \]

Step 5: Calculate probability for additional three years

To find the conditional probability that the organism will live another three years given it has survived the first three years, use the formula:\[ P(X > 6 | X > 3) = \frac{S(6)}{S(3)} \] Where \( S(x) = e^{-\frac{0.04}{4.1} x^{4.1}} \). Calculate \( S(6) \):\[ S(6) = e^{-\frac{0.04}{4.1} \times 6^{4.1}} \] Plug the numbers and calculate:\[ S(6) \approx e^{-5.4527} \approx 0.0043 \] Now calculate the conditional probability:\[ \frac{S(6)}{S(3)} = \frac{0.0043}{0.4513} \approx 0.0095 \]

Key Concepts

Hazard Rate Function

The hazard rate function, also known simply as the hazard function, is an essential component of survival analysis. This function, \( \lambda(x) \), characterizes the instantaneous risk of an event happening at a particular time, given that the event hasn't occurred yet. In our case, the event in question is the death of an organism. The provided hazard rate function for this problem is given as \( \lambda(x)=0.04 x^{3.1} \). This specific form tells us that the risk of the organism dying increases with age in a curvilinear manner.

• The hazard rate is crucial to understand because it helps in calculating the survival probability of an organism.
• The function typically depends on the age or time variable \( x \) and expresses how the risk of dying varies over time.

The higher the value of \( \lambda(x) \), the higher the risk of the event happening soon. Here the function includes a power of \( x \), suggesting a non-linear increase in risk as time progresses. Understanding this function is crucial to further calculations of survival and conditional probabilities.

Survival Function

In survival analysis, the survival function, denoted \( S(x) \), represents the probability that an organism, or any entity, survives beyond a given time \( x \). The expression for the survival function is given through the hazard rate function as:\[ S(x) = e^{-\int_0^x \lambda(t) \, dt} \]Let's break it down:

• The integral of the hazard rate function from time \( 0 \) to \( x \) is calculated. This accumulates the risk over time.
• The negative of this integral is the exponent in the exponential function, which helps compute the survival probability.

In our example, the survival function derived is\[ S(x) = e^{-\frac{0.04}{4.1} x^{4.1}} \]which allows us to compute the probability of the organism living beyond specific time points. It's key to note that as \( x \) increases, \( S(x) \) tends to decrease, reflecting higher cumulative risk.

Conditional Probability

Conditional probability is a concept that helps us determine the probability of an event occurring given that another event has already occurred. In survival analysis, it's often used to calculate the likelihood an organism survives additional time after surviving some initial period.Given the survival function \( S(x) \), the conditional probability is calculated as:\[ P(X > a + b \mid X > a) = \frac{S(a+b)}{S(a)} \]Let's look at our problem: We want to know the probability that the organism will live another three years given it has already lived the first three years:

• Calculate \( S(6) \) for six years.
• Calculate \( S(3) \) for three years.
• The conditional probability is then the ratio \( \frac{S(6)}{S(3)} \).

Using this approach, we find that the probability of living another three years, given the organism has already survived three years, is approximately 0.0095. This low probability indicates significant risk associated with already having reached this age.