If X is a Poisson variate with
`P(X=2)=(2)/(3)P(X=1)`, find `P(X=0)` and `P(X=3)`

Answer

To solve the problem, we need to find \( P(X=0) \) and \( P(X=3) \) given that \( P(X=2) = \frac{2}{3} P(X=1) \) for a Poisson random variable \( X \) with parameter \( \lambda \).

### Step-by-Step Solution:

1. **Understand the Poisson Probability Formula**:
The probability mass function for a Poisson random variable is given by:
\[ P(X = k) = \frac{e^{-\lambda} \lambda^k}{k!} \]
where \( k \) is a non-negative integer.

2. **Set Up the Given Equation**:
According to the problem, we have:
\[ P(X=2) = \frac{2}{3} P(X=1) \]
Using the Poisson formula, we can express this as:
\[ \frac{e^{-\lambda} \lambda^2}{2!} = \frac{2}{3} \cdot \frac{e^{-\lambda} \lambda^1}{1!} \]

3. **Simplify the Equation**:
Cancel \( e^{-\lambda} \) from both sides (since it is positive) and substitute the factorial values:
\[ \frac{\lambda^2}{2} = \frac{2}{3} \cdot \lambda \]

4. **Multiply Both Sides by 6**:
To eliminate the fractions, multiply both sides by 6:
\[ 3\lambda^2 = 4\lambda \]

5. **Rearrange the Equation**:
Rearranging gives us:
\[ 3\lambda^2 - 4\lambda = 0 \]
Factor out \( \lambda \):
\[ \lambda(3\lambda - 4) = 0 \]
This gives us two solutions: \( \lambda = 0 \) or \( \lambda = \frac{4}{3} \). Since \( \lambda = 0 \) is not a valid parameter for a Poisson distribution, we take:
\[ \lambda = \frac{4}{3} \]

6. **Calculate \( P(X=0) \)**:
Now we can find \( P(X=0) \):
\[ P(X=0) = \frac{e^{-\lambda} \lambda^0}{0!} = e^{-\frac{4}{3}} \cdot 1 = e^{-\frac{4}{3}} \]
Approximating \( e^{-\frac{4}{3}} \):
\[ P(X=0) \approx 0.264 \]

7. **Calculate \( P(X=3) \)**:
Now we find \( P(X=3) \):
\[ P(X=3) = \frac{e^{-\lambda} \lambda^3}{3!} = \frac{e^{-\frac{4}{3}} \left(\frac{4}{3}\right)^3}{6} \]
Calculate \( \left(\frac{4}{3}\right)^3 = \frac{64}{27} \):
\[ P(X=3) = \frac{e^{-\frac{4}{3}} \cdot \frac{64}{27}}{6} = \frac{64 e^{-\frac{4}{3}}}{162} \]
Approximating gives:
\[ P(X=3) \approx 0.104 \]

### Final Answers:
- \( P(X=0) \approx 0.264 \)
- \( P(X=3) \approx 0.104 \)