. The function 𝑓(𝑡) = {
3 × 10−9
𝑡
2
(100 − 𝑡)
2
; 0 ≤ 𝑡 ≤ 100
0; 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
. If the probability that a
person will die in the time interval (𝑡1,𝑡2) is given by 𝑃(𝑡1 ≤ 𝑡 ≤ 𝑡2
) = ∫ 𝑓(𝑡)𝑑𝑡 𝑡2
𝑡1
.
Determine (i) the probability that a person will die between the ages 60 & 70. (ii) the
probability that he will die between those ages, assuming he lived upto 60

We are given the probability density function (PDF) \( f(t) \) of the time of death \( t \), and we need to determine the probability that a person will die between the ages of 60 and 70. Additionally, we want to determine the conditional probability that a person dies between these ages given they have already lived up to 60.

### Given:
\[
f(t) =
\begin{cases}
3 \times 10^{-9} \cdot t^2 \cdot (100 - t)^2 & \text{if } 0 \leq t \leq 100 \\
0 & \text{otherwise}
\end{cases}
\]
The probability that a person dies in the time interval \( (t_1, t_2) \) is given by:
\[
P(t_1 \leq t \leq t_2) = \int_{t_1}^{t_2} f(t) \, dt
\]

### Part (i) – Probability of dying between the ages 60 and 70:

We want to calculate:
\[
P(60 \leq t \leq 70) = \int_{60}^{70} f(t) \, dt
\]

### Part (ii) – Conditional probability of dying between ages 60 and 70, given survival to 60:

This is a conditional probability, and it is given by:
\[
P(60 \leq t \leq 70 \mid t \geq 60) = \frac{P(60 \leq t \leq 70)}{P(t \geq 60)}
\]
Where \( P(t \geq 60) = 1 - P(0 \leq t \leq 60) = 1 - \int_{0}^{60} f(t) \, dt \).

I'll now proceed with the integration to find the required probabilities.

Let's first calculate the probability for part (i).### Results:

(i) The probability that a person will die between the ages of 60 and 70 is approximately \( 0.154 \) (15.4%).

(ii) The conditional probability that a person will die between the ages of 60 and 70, given that they have already lived up to 60, is approximately \( 0.486 \) (48.6%).

Would you like more details or further clarification on any part of the process?

Here are 5 related questions to expand your understanding:
1. How is the probability density function (PDF) normalized over its domain?
2. What is the significance of conditional probability in survival analysis?
3. How do we interpret the integral of a PDF over a certain interval?
4. What happens to the probabilities if we change the range of ages, say from 70 to 80?
5. How does survival probability vary with different functional forms of \( f(t) \)?

**Tip**: Conditional probability is often used in survival analysis to make predictions given partial information about an individual's survival up to a certain point.

The function f(t) = 3 × 10^−9 t^2 (100 − t)^2; 0 ≤ t ≤ 100, 0 otherwise. If the probability that a person will die in the time interval (t1, t2) is given by P(t1 ≤ t ≤ t2) = ∫ f(t) dt from t1 to t2, determine (i) the probability that a person will die between the ages 60 & 70, (ii) the probability that he will die between those ages, assuming he lived up to 60.

Learn how to calculate the probability that a person will die between the ages of 60 and 70 using a probability density function (PDF) and integration. This problem also covers conditional probability assuming survival up to age 60. Suitable for students at an undergraduate level, covering key concepts in calculus and survival analysis.

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