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SELESAIKAN SOAL TERSEBUT SESUAI DENGAN TEORI AKTUARIA/ PROSES STOKASTIK/ TEORI RESIKO

The problem provided involves insurance claims modeled by a Poisson process, which is a common technique in actuarial science, stochastic processes, and risk theory. Let's break down and solve the problem step by step.

### Key Points from the Problem:
- There are two types of insurance policies: **Policy A** and **Policy B**.
- Claims follow a **Poisson process** with a rate parameter (λ) of **9 claims per day**.
- The probability that a randomly chosen claim is from **Policy A** is \( \frac{1}{3} \).
- We are asked to calculate:
  1. The probability that **claims from Policy A (or B)** on a given day are less than 2.
  2. The probability that **total claims** on a given day are less than 2.

### 1. Probability for Claims from Policy A (or B) Being Less Than 2

Since the claims follow a Poisson process, the number of claims per day \( N \) is modeled by a Poisson random variable:

\[
P(N = n) = \frac{\lambda^n e^{-\lambda}}{n!}
\]

For **Policy A**:
- The probability that a claim is from **Policy A** is \( P(A) = \frac{1}{3} \).
- The rate of claims for Policy A is \( \lambda_A = P(A) \times \lambda = \frac{1}{3} \times 9 = 3 \).

The Poisson distribution for Policy A with \( \lambda_A = 3 \):

\[
P(N_A = n) = \frac{3^n e^{-3}}{n!}
\]

We need to calculate \( P(N_A < 2) \), which means we sum the probabilities for 0 and 1 claims:

\[
P(N_A < 2) = P(N_A = 0) + P(N_A = 1)
\]

\[
P(N_A = 0) = \frac{3^0 e^{-3}}{0!} = e^{-3}
\]
\[
P(N_A = 1) = \frac{3^1 e^{-3}}{1!} = 3e^{-3}
\]

Thus:

\[
P(N_A < 2) = e^{-3} + 3e^{-3} = 4e^{-3}
\]

Using the approximation \( e^{-3} \approx 0.0498 \):

\[
P(N_A < 2) \approx 4 \times 0.0498 = 0.1992
\]

Therefore, the probability that claims for **Policy A** are less than 2 is approximately **0.199**.

For **Policy B**:
- Since the total claim rate is 9, and Policy A accounts for \( \lambda_A = 3 \), the claim rate for Policy B is \( \lambda_B = 9 - 3 = 6 \).
- Similar calculations apply for Policy B, and you could calculate \( P(N_B < 2) \) using \( \lambda_B = 6 \).

### 2. Probability that Total Claims on a Given Day are Less Than 2

For total claims, we use the Poisson distribution with \( \lambda = 9 \). We are asked to find \( P(N < 2) \), which again means calculating for 0 and 1 claim:

\[
P(N = 0) = \frac{9^0 e^{-9}}{0!} = e^{-9}
\]
\[
P(N = 1) = \frac{9^1 e^{-9}}{1!} = 9e^{-9}
\]

Thus:

\[
P(N < 2) = P(N = 0) + P(N = 1) = e^{-9} + 9e^{-9} = 10e^{-9}
\]

Using the approximation \( e^{-9} \approx 0.000123 \):

\[
P(N < 2) \approx 10 \times 0.000123 = 0.00123
\]

Therefore, the probability that total claims are less than 2 is approximately **0.00123**.

### Summary of Solutions:
1. The probability that **claims from Policy A** are less than 2: **0.199**.
2. The probability that the **total claims** on a given day are less than 2: **0.00123**.

Would you like further details or have any specific questions?

### Related Questions:
1. How does the Poisson distribution apply to insurance claims modeling?
2. What is the significance of the parameter λ in the Poisson process?
3. How do you approximate exponential values like \( e^{-3} \) and \( e^{-9} \) in calculations?
4. How can you calculate the expected number of claims for both policies combined?
5. How do stochastic processes relate to risk theory in actuarial science?

### Tip:
When dealing with Poisson distributions, summing probabilities for small ranges (like 0 or 1 claims) can be efficiently done using exact formulas, but for large λ, normal approximations may be useful!

A company has two insurance policies, A and B. Claims are submitted following a Poisson process with a parameter of 9 claims per day. The probability that a claim is from Policy A is 1/3. Calculate: (1) The probability that claims from Policy A (or B) on a given day are fewer than 2. (2) The probability that the total number of claims submitted in a day is fewer than 2.

This article solves an actuarial problem involving a Poisson process where an insurance company with two policies, A and B, receives claims. The claims follow a Poisson distribution with a rate of 9 per day. Learn how to calculate the probability that claims from Policy A or B, or the total claims, are fewer than 2.

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