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The exponential distribution is a continuous probability distribution often used to model the time or space between events in a Poisson process. It is characterized by a parameter \( \lambda \), known as the rate parameter. This parameter helps determine how quickly an event occurs. The exponential distribution is particularly useful for modeling scenarios where things "wear out" or "fail," like the lifespan of a lightbulb.
Understanding the basic properties of this distribution is key:

• The mean of an exponentially distributed random variable \( X \) is \( E(X) = \frac{1}{\lambda} \).
• The variance is \( \text{Var}(X) = \frac{1}{\lambda^2} \).
• The probability that the event happens before time \( t \) is given by the cumulative distribution function (CDF): \( P(X < t) = 1 - e^{-\lambda t} \).

In the context of lightbulbs, an exponential distribution helps us determine the probability of failure within a certain time, considering each lightbulb independently of others.

The binomial distribution is a discreet probability distribution of the number of successes in a sequence of \( n \) independent experiments or trials, each asking a yes-no question. Such questions can either result in "success" or "failure." The distribution is defined by two parameters: \( n \), the number of trials, and \( p \), the probability of success on each trial.
The formula for finding the probability of exactly \( k \) successes (failures in our exercise with lightbulbs) is:

• \( P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \)

This is where \( \binom{n}{k} \) is a combination that represents the number of ways to choose \( k \) successes from \( n \) trials.
In our exercise, when calculating the number of bulbs that fail before time \( t \), we use the cumulative probability that a single bulb fails before time \( t \) as our "success" probability \( p \), while \( n \) stands for the ten bulbs collectively being tested.

In probability theory, events are defined as independent if the occurrence of one event does not affect the likelihood of the occurrence of another. This concept simplifies the calculation of probabilities in complex scenarios since you can treat each event separately.
In mathematical terms, two events \( A \) and \( B \) are independent if:

• \( P(A \cap B) = P(A) \cdot P(B) \)

For our context, when calculating the probability of independent lightbulbs failing, knowing the result of one bulb's lifetime does not impact or inform us about others. This gives us the freedom to consider each bulb's failure individually when computing probabilities, such as all bulbs failing before a given time \( r \).
Recognizing the independence of events allows us to combine individual probabilities to find the total probability in a straightforward manner.