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The Cumulative Distribution Function (CDF) provides a comprehensive way to express the probability of a random variable being less than or equal to a certain value. For the random variable \(X\), the CDF, denoted as \(F(x)\), is calculated as \(F(x) = P(X \leq x)\). The CDF starts at zero and increases to one, providing a full picture of how probability accumulates over various outcomes.
This function is particularly useful for finding the probability over a range of outcomes, as it essentially adds up the probabilities for each outcome up to a certain point.
In the context of our problem, the CDF is calculated by summing up the probabilities from the Probability Mass Function (PMF) for each possible number of late arrivals up to a particular number.
This is instrumental in determining the probability that the number of late arrivals is between two values, such as calculating \(P(2 \leq X \leq 6)\) by evaluating \(F(6) - F(1)\).

A Bernoulli random variable is a simple type of discrete random variable that can take on only two possible outcomes: success or failure. These outcomes are typically encoded as 1 and 0, respectively. Each trial in a Bernoulli process has a probability \(p\) of resulting in a success and \(1-p\) of resulting in failure.
In our exercise, each couple or individual has a probability of 0.4 of being late, which fits the Bernoulli model since there are only two outcomes: being late (success) or being on time (failure).
When we sum independent Bernoulli random variables, as with multiple individuals or couples deciding to attend a seminar, we get a Binomial distribution. This concept is key to understanding how the individual late arrivals translate into the total number of late attendees, which is represented by the random variable \(X\). By considering each couple or individual as a Bernoulli trial with \(p = 0.4\), the overall behavior of \(X\) can be understood using Binomial distribution principles.

A Binomial Distribution is a statistical distribution that arises from performing a fixed number of independent Bernoulli trials, each with the same probability of success. It is fully characterized by two parameters: \(n\), the number of trials, and \(p\), the probability of success on each trial. In the context of the exercise, the couples and individuals attending the seminar can be considered as separate groups of Bernoulli trials.
We have the following:

• Couples: 3 trials, each with probability 0.4 of being late, resulting in \(Y \sim \text{Binomial}(3, 0.4)\).
• Individuals: 2 trials, also with 0.4 probability of being late, leading to \(Z \sim \text{Binomial}(2, 0.4)\).

The probability mass function for a binomial distribution \(P(X = k)\) is given by \(\binom{n}{k} p^k (1-p)^{n-k}\), where \(\binom{n}{k}\) is the binomial coefficient. This helps us compute the likelihood of any number \(k\) of successes across the trials.
For example, in determining the PMF of \(X\), our task involves summing up scenarios from these binomial distributions to represent all possible outcomes where a given number of people arrive late.

In probability theory, independent events are those whose occurrence or non-occurrence does not affect the occurrence of another event. It means that knowing the outcome of one event gives no information about the outcome of another.
This principle is crucial when calculating probabilities involving multiple events, as it significantly simplifies the math involved.In the given problem, the late arrival of each couple or individual is considered to be independent of others.
This assumption allows us to multiply probabilities across events to find joint probabilities. For example, the probability that a specific combination of couples and individuals arrive late can be computed by multiplying their respective lateness probabilities (0.4 for each).
The independence assumption simplifies the modeling of our random variable \(X\), as it assures that the Binomial models for the groups can be used without additional dependence terms. As the problem proceeds through calculations, this concept facilitates finding exact probabilities that would be much more complex otherwise.