Bits are transmitted through a digital transmission system but errors might occur
(the probability of an error is ϵ). We let t be the number of bits that were transmitted
until an error occured.
(a) What is the probability of p(t | ϵ)?

To find the probability \( p(t \mid \epsilon) \), we need to think of the problem in terms of a **geometric distribution**. The situation involves transmitting bits through a system until the first error occurs, and the probability of an error occurring at any given transmission is \( \epsilon \).

### Step-by-Step Breakdown:

- Let \( t \) be the number of bits transmitted until the first error occurs.
- The probability of an error occurring is \( \epsilon \).
- The probability of a successful transmission (no error) is \( 1 - \epsilon \).

This setup is modeled by a **geometric distribution**, which describes the number of trials needed to get the first success (in this case, the first error). The probability mass function (PMF) of a geometric distribution is given by:

\[
p(t \mid \epsilon) = (1 - \epsilon)^{t-1} \cdot \epsilon
\]

### Explanation of the formula:
- \( (1 - \epsilon)^{t-1} \): The probability that the first \( t-1 \) bits are transmitted successfully (no errors).
- \( \epsilon \): The probability that the \( t \)-th bit is in error.

Thus, the probability of the first error occurring on the \( t \)-th bit is:

\[
p(t \mid \epsilon) = (1 - \epsilon)^{t-1} \cdot \epsilon
\]

This is the answer to part (a).

### Summary:
- The number of bits \( t \) until the first error occurs follows a geometric distribution with parameter \( \epsilon \).
- The probability mass function is \( p(t \mid \epsilon) = (1 - \epsilon)^{t-1} \cdot \epsilon \).

Do you want more details or have any questions?

Here are 5 related questions to expand on this topic:
1. How would the expectation \( E[t] \) for the number of bits transmitted before the first error be calculated?
2. What is the variance of \( t \) in this geometric distribution?
3. How does the probability change if the error probability \( \epsilon \) is very small?
4. How can we extend this model to account for multiple errors (i.e., find the distribution of the time until the second error)?
5. How would you calculate the probability of no errors occurring in the first \( t \) transmissions?

**Tip:** Geometric distributions are memoryless, meaning the probability of an error happening

Bits are transmitted through a digital transmission system but errors might occur (the probability of an error is ϵ). We let t be the number of bits that were transmitted until an error occured. (a) What is the probability of p(t | ϵ)?

Understand how to calculate the probability of the first error occurring at the t-th bit in a digital transmission system using geometric distribution. This solution involves key concepts in probability theory and provides a detailed breakdown of the formula p(t | ϵ) = (1 - ϵ)^(t-1) * ϵ. Suitable for undergraduate-level students.

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