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Discrete Distributions

Discrete Uniform Distribution

If a random variable assumes the values with equal probability, then follows a discrete uniform distribution.

The probability mass function for is then given

Misplaced &\frac{1}{k} & x = $x_{1},..., x_k$ \\ 0 & otherwise \end{cases}

Thus, if follows the discrete uniform distribution as mentioned above,

Theorem 2 X follows the discrete uniform distribution with R_{X}= \{x_{1}, .. , x_k\}

The expectation of is then given:

The variance of is then given:

Bernoulli Trials

Bernoulli trial

A random experiment with only two possible outcomes.

The outcomes are generally coded:

  • 0: failure
  • 1: success

Bernoulli Random Variable

Let be the number of success in a Bernoulli trial. Then has only two possible values, 1 or 0, and is called a Bernoulli random variable.

Let denote the probability of success for a Bernoulli trial.

Then, has the probability mass function:

This can also be written:

Notation

We denote a Bernoulli random variable by , and . Then, the probability mass function becomes:

Theorem 5

For a Bernoulli random variable ,

In certain instances, may rely on one or more unknown quantities - where different values of the quantities lead to different probability distributions. These quantities are called parameters.

Thus, for the Bernoulli distribution, the parameter is .

Bernoulli Process

A sequence of repeatedly performed independent and identical Bernoulli trials.

Generates a sequence of independent and identically distributed Bernoulli random variables.

Binomial Distribution

Binomial Random Variable

A Binomial random variable counts the number of successes in trials of a Bernoulli Process.

Suppose we have trials where:

  • probability of success for each trial is
  • trials are independent.

Then, the number of successes denoted by has a binomial distribution .

The probability of getting exactly successes is given as:

It can be shown that

Remark

When , probability mass function for the binomial random variable is reduced to:

# computing binoms in R
pbinom(x, n, p) #P(X <= x);
pbinom(x, n, p, lower.tail = F) #P(X > x);
dbinom(x, n, p) #P(X = x)

Negative Binomial Distribution

Consider a Bernoulli process, where the variable of interest is the number of trials needed so that number of successes occur.

Negative Binomial Distribution

The random variable is defined as the number of independent and identically distributed Bernoulli() trials needed until the th success occurs.

Then, follows a Negative Binomial distribution, denoted by .

The probability mass function of is given by:

Then,

The derivation of the of can be seen as follows:

The event refers to the event where trials is needed to get successes. This event can be split into when

  • There are successes in the first trials (define this event )
  • There is a success on the trial. (define this event )

Then,

# computing negative binoms in R
dnbinom(x - k, k, p) #P(X = x);
pnbinom(x - k, k, p) #P(X <= x);
pnbinom(x - k, k, p) #P(X > x);

Geometric Distribution

Consider a Bernoulli process where the random variable of interest is the amount of Bernoulli () trials needed until first success occurs.

Geometric distribution

Let be the number of independent and identically distributed Bernoulli() trials needed until the first success occurs. Then follows a Geometric distribution, denote by

The probability mass function of is given by: It then can be shown that:

Poisson Distribution

Poisson random variable

The Poisson random variable denotes the number of events occuring in a fixed period of time or fixed region.

is used to denote the distribution, where is the expected number of occurences during given period/region.

The probability mass function is:

Poisson process

A continuous time process, where the number of occurences within a given interval of time is counted.

The defining properties with rate parameter

  • the expected number of occurences in an interval of length T is
  • no simultaneous occurences
  • number of occurences in disjoint time intervals are independent

The number of occurences follow a distribution.

Poisson Approximation to Binomial

The Poisson random variable can be used for approximation of binomial random variable under certain conditions.

Poisson Approximation to Binomial

. Suppose that such that remains a constant. Approximately, .

Good approximation: n \geq 20, p \leq 0.05 or n \geq 100, np \leq 10

# computing poisson probabilities
 
dpois(x, lambda) # P(X = x)
ppois(x, lambda) # P(X <= x)
ppois(x, lambda, lower.tail = F) # P(X > x)

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