Suppose within a 10 year span, you want to check how many bulbs have already malfunctioned. Lets call it X

On average how many bulbs do you think have malfunctioned in the timeframe or what is E(X)?

What is the pmf of X?

To analyze the problem of malfunctioning bulbs over a 10-year span, we need to use probability theory. Let's break it down:

### 1. Assumptions and Definitions:

- **X**: The random variable representing the number of bulbs that have malfunctioned in a 10-year period.
- We assume that the malfunctions of the bulbs follow a Poisson process because this process is commonly used for modeling random events that occur independently over time.
- If the malfunctions are assumed to occur at an average rate \( \lambda \) per year, then over 10 years, the average rate would be \( 10\lambda \).

### 2. Expected Value, E(X):

Since \( X \) follows a Poisson distribution:

\[
X \sim \text{Poisson}(\lambda t)
\]

where \( t = 10 \) years, the expected number of malfunctions \( E(X) \) is given by:

\[
E(X) = \lambda t = 10\lambda.
\]

Therefore, the average number of bulbs that have malfunctioned in 10 years is \( 10\lambda \).

### 3. Probability Mass Function (PMF) of \( X \):

The probability mass function (PMF) of a Poisson-distributed random variable \( X \) with parameter \( \lambda t \) is given by:

\[
P(X = k) = \frac{(\lambda t)^k e^{-\lambda t}}{k!}, \quad \text{for } k = 0, 1, 2, \ldots
\]

Substituting \( t = 10 \), we get:

\[
P(X = k) = \frac{(10\lambda)^k e^{-10\lambda}}{k!}, \quad \text{for } k = 0, 1, 2, \ldots
\]

This is the PMF of \( X \), where \( k \) represents the number of bulbs that have malfunctioned.

### Summary:

1. **Expected Number of Malfunctions (E(X))**: \( 10\lambda \).
2. **PMF of X**: \( P(X = k) = \frac{(10\lambda)^k e^{-10\lambda}}{k!} \).

Would you like further details or have any questions about this analysis?

### Related Questions:

1. How would the PMF change if the rate of malfunction \( \lambda \) is not constant over time?
2. What is the variance of the number of malfunctions, \( \text{Var}(X) \), for a Poisson distribution?
3. How would you model the malfunctions if they are not independent over time?
4. How can you compute the probability that fewer than 5 bulbs malfunction in the 10-year span?
5. How would you estimate \( \lambda \) if you have data on the number of malfunctions over several 10-year periods?

### Tip:

When modeling real-life situations with probability distributions, ensure that the assumptions of the model align well with the actual characteristics of the data or phenomenon.

Explore the analysis of bulb malfunction probabilities over a 10-year span using probability theory and the Poisson distribution. Learn about the expected number of malfunctions, E(X), and the probability mass function (PMF) of X. Suitable for advanced high school students and beyond.

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