91Ó°ÊÓ

A Bernoulli distribution is one of the simplest forms of probability distribution and is used to represent a single experiment that has two possible outcomes, typically labeled as success and failure. The Bernoulli distribution is characterized by a single parameter, \( p \), which represents the probability of success.

• If \( X \) is a random variable with a Bernoulli distribution and \( p \) is the probability of success, then \( \mathrm{P}(X = 1) = p \) and \( \mathrm{P}(X = 0) = 1 - p \).
• The expected value or mean of a Bernoulli distribution is \( p \), and the variance is \( p(1-p) \).

In the context of our problem, we have two independent random variables, \( X \) and \( Y \), each following a Bernoulli distribution. \( X \) has a parameter \( p = \frac{1}{2} \), meaning there is an equal chance of success and failure. Meanwhile, \( Y \) has a parameter \( q = \frac{1}{4} \), indicating a lower probability of success compared to failure.

The binomial distribution is a more complex distribution that deals with the number of successes in a fixed number of independent Bernoulli trials. It is defined by two parameters: \( n \), the number of trials, and \( r \), the probability of success in a single trial.

• A random variable \( Z \) that follows a binomial distribution is denoted as \( \operatorname{Bin}(n, r) \).
• The probability of obtaining exactly \( k \) successes in \( n \) trials is given by the formula \[ \mathrm{P}(Z = k) = \binom{n}{k} r^k (1-r)^{n-k}, \] where \( \binom{n}{k} \) is the binomial coefficient.

In our exercise, it's proposed that \( X+Y \) should follow a binomial distribution when \( p = q = r \). However, since \( p eq q \) in our case, where \( p = \frac{1}{2} \) and \( q = \frac{1}{4} \), the resulting distribution of \( X+Y \) does not fit the binomial model as the conditions are not met. This shows a distinctive property as the sum of independent Bernoulli random variables with different parameters doesn't form a binomial distribution.

Random variables \( X \) and \( Y \) are said to be independent if the occurrence of an event related to \( X \) does not affect the probability of an event related to \( Y \) and vice versa. This concept is crucial when considering combinations of random variables.

• For independent random variables, the probability of simultaneous events can be determined by multiplying their individual probabilities: \( \mathrm{P}(X = x \text{ and } Y = y) = \mathrm{P}(X = x) \times \mathrm{P}(Y = y) \).
• Independence is key when dealing with the sum or product of random variables, as it allows simplification in calculations due to non-interaction of events.

In the given exercise, the independence of \( X \) and \( Y \) allows us to compute the probabilities \( \mathrm{P}(X+Y = k) \) by multiplying the respective probabilities of event outcomes for \( X \) and \( Y \). This makes it possible to separately calculate and then sum up the distinct probabilities needed to analyze the distribution of \( X + Y \).