Question 1194139: X and Y sizes are independent, X ~ P(3), Y ~ G(0.23). Find the probability P(Y=2x)
Answer by parmen(42)   (Show Source):
You can put this solution on YOUR website! **1. Define the Distributions**
* **X ~ Poisson(λ = 3)**:
- Probability Mass Function (PMF): P(X = x) = (e^(-λ) * λ^x) / x!
- Where λ is the rate parameter (here, λ = 3) and x is the number of occurrences.
* **Y ~ Geometric(p = 0.23)**:
- Probability Mass Function (PMF): P(Y = y) = (1-p)^(y-1) * p
- Where p is the probability of success (here, p = 0.23) and y is the number of trials until the first success.
**2. Calculate P(Y = 2x)**
* Since X and Y are independent, we can calculate the joint probability as:
P(Y = 2x) = P(X = x) * P(Y = 2x)
* We need to sum this probability over all possible values of x:
P(Y = 2x) = Σ [P(X = x) * P(Y = 2x)] for x = 0, 1, 2, ...
* **Note:** The geometric distribution is defined for y ≥ 1. Therefore, we need to consider only values of x where 2x ≥ 1, which means x ≥ 0.5. Since x is a discrete variable, we start with x = 1.
**3. Implementation**
* We can implement this calculation using Python:
```python
import numpy as np
from scipy.stats import poisson, geom
lambda_x = 3
p_y = 0.23
prob_y_2x = 0
for x in range(1, 10): # We can limit the range to 10 for practical purposes
prob_x = poisson.pmf(x, mu=lambda_x)
prob_y = geom.pmf(2*x, p=p_y)
prob_y_2x += prob_x * prob_y
print(f"P(Y=2x) = {prob_y_2x}")
```
**Result:**
This Python code will give you the approximate value of P(Y = 2x).
**Important Note:**
* The geometric distribution is defined for discrete values of y ≥ 1.
* The Poisson distribution can take on non-negative integer values (0, 1, 2, ...).
* The condition Y = 2x might not always be satisfied for all possible values of X and Y, as the relationship between the two random variables might not always hold true.
This calculation provides an approach to finding the probability P(Y = 2x) given the specified distributions.
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