The Exponential Distribution

0.1.1.1 Binomial Experiment

A binomial experiment (also known as a Bernoulli trial) is a statistical experiment that has the following properties:

• The experiment consists of \(n\) repeated trials.
• Each trial can result in just two possible outcomes. We call one of these outcomes a success and the other, a failure.
• The probability of success, denoted by \(p\), is the same on every trial.
• The trials are independent; that is, the outcome on one trial does not affect the outcome on other trials.

Consider the following statistical experiment. You flip a coin five times and count the number of times the coin lands on heads. This is a binomial experiment because:

• The experiment consists of repeated trials. We flip a coin five times.
• Each trial can result in just two possible outcomes : heads or tails.
• The probability of success is constant : 0.5 on every trial.
• The trials are independent; that is, getting heads on one trial does not affect whether we get heads on other trials.

• A binomial experiment with n trials and probability \(p\) of success will be denoted by \[B(n, p)\]
• Frequently, we are interested in the in a binomial experiment, not in the order in which they occur.
• Furthermore, we are interested in the probability of that number of successes.

The probability of exactly k successes in a binomial experiment B(n, p) is given by \[ P(X=k) = P(k \mbox{ successes }) = \;^nC_k \times p^{k} \times (1-p)^{n-k}\]

• X: Discrete random variable for the number of successes (variable name) *\(k\) : Number of successes (numeric value)

• \(P(X=k)\) ``probability that the number of success is \(k\)“.

• \(n\) : number of independent trials

• \(p\) : probability of a success in any of the \(n\) trial.

• \(1-p\) : probability of a failure in any of the \(n\) trial.

0.1.1.2 Binomial Distribution : Probability Density Function}

A binomial experiment is one that possesses the following properties:

• The experiment consists of n repeated trials;

• Each trial results in an outcome that may be classified as a success or a failure (hence the name, binomial);

• The probability of a success, denoted by p, remains constant from trial to trial and repeated trials are independent.

  The number of successes X in n trials of a binomial experiment is called a .

0.1.1.3 Binomial Probability Formula

The probability of exactly k successes in a binomial experiment \(Bin(n, p)\) is given by

\[ P(X=k) = P(k \mbox{ successes in ``n" trails }) = \;^nC_k \times p^{k} \times (1-p)^{n-k}\]

• X: Discrete random variable for the number of successes (variable name)

• \(k\) : Number of successes (numeric value)

• k= 0, 1, 2, … n

• \(P(X=k)\) ``probability that the number of success is \(k\)“.

• \(n\) : number of independent trials

• \(p\) : probability of a success in any of the \(n\) trial.

• \(1-p\) : probability of a failure in any of the \(n\) trial.

• \({^nC_k}\) is a combination value, found using the Choose operator.

0.1.2.1 Expectation and Variance

If the random variable X has a binomial distribution with parameters \(n\) and \(p\), we write \[ X \sim Bin(n,p) \] Only these two parameters are needed to determine the probability of an event.

• The expected value of \(X\) is: \[\operatorname{E(X)} = n \times p \]

• The variance of \(X\) is: \[\operatorname{Var(X)} = n \times p \times (1-p) = n\times p \times q \]

• \(p\) is the probability of success * \(q\) is the probability of failure in a binomial trial

• \(n\) is the number of independent trials

Interpretation: If \(n=100\), and \(p=0.25\), then the average number of successes will be 25.