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In probability theory, a probability distribution describes how the probabilities are distributed over the outcomes of a random variable. In our exercise, we are examining the distortion of bits during digital information transmission. Each bit can independently have high, moderate, or low distortion, with respective probabilities of 0.01, 0.04, and 0.95.

For example, to understand the distribution of a random variable like the number of bits with high distortion, we employ a probability distribution. This will tell us how likely each number of high-distortion bits is when we send three bits. The probability distribution can be expressed using mathematical functions or tables, and these probabilities must add up to 1.

• Important Concept: The sum of probabilities in a probability distribution across all possible outcomes should equal 1. This ensures that all potential events are accounted for in the model.

Understanding probability distributions helps us make predictions and informed decisions by knowing what outcomes are more likely under given conditions.

The binomial distribution is a specific type of probability distribution that applies when there are a fixed number of independent trials, each with two possible outcomes, like "success" and "failure". In our context, success might refer to a bit experiencing high distortion.

In the given problem, we are said to transmit three bits, each possibly experiencing high distortion with a probability of 0.01. This situation is modeled appropriately using a binomial distribution, where:

• The number of trials ( n ) is 3.
• The probability of success ( p ) is 0.01.

We compute probabilities for different numbers of successes using the binomial probability formula: \[P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}\]

• Where:
  • \(k\) is the number of successes.
  • \(\binom{n}{k}\) represents the number of combinations.

The mean and variance are found directly using:

• Mean ( \(\mu\) ): \(np\)
• Variance ( \(\sigma^2\) ): \(np(1-p)\)

This distribution is crucial for managing binary outcomes and understanding scenarios like those in the transmission example.

Conditional probability is the probability of an event occurring given that another event has already occurred. It is especially useful for understanding how information or outcomes can influence each other, which is key in decision-making processes and data analysis.

In the problem at hand, we are asked to find the probability distribution of the number of high distortion bits (denoted by X), given that two bits have moderate distortion (denoted by Y = 2). The formula for conditional probability is:\[P(A|B) = \frac{P(A \cap B)}{P(B)}\]

• Where:
  • \(A\) is the event of interest (e.g., number of high distortion bits).
  • \(B\) is the given condition (e.g., Y = 2).

When dealing with conditional probability, understanding that the context changes based on existing information is vital. This means that in our example, since there are already two bits with moderate distortion, the only possible outcome for X must adjust according to this prior knowledge, leading to the result that X can only equal 0 given Y=2. This eliminates any variability, thus making the conditional variance 0. Such deductions highlight how conditional probability can refine our predictions or expectations in response to new data.