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Standard Deviation of a Normal Distribution

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Types of Distributions
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distribution is a description of the possible values of a random variable and the possible occurrences of these values.

A probability distribution is a graph or table that specifies the probablity associated with each possible value that the random variable can assume.

  • Mean Value or Expected Value

    The mean value, or expected value , of a discrete random variable X is given by the following equation:

    \mu=E(x)=\sum_{} xp(x)

  • Variance

    The variance of a discrete random variable is given by the following formula:

    \sigma^2 = \sum_{} (x-\mu)^2 P(x)

  • Standard Deviation

    The square root of the variance, or, in other words, the square root of \sigma^2 , is the standard deviation of a discrete random variable:

    \sigma=\sqrt{\sigma^2}

A sampling distribution is the probability distribution of a given statistic based on a random sample.

Normal Distribution

A normal distribution is called a bell curve because its shape is comparable to a bell. It has this shape because the majority of the data is concentrated at the middle and slowly decreases symmetrically on either side.  A normal distribution can be described by the mean and standard deviation of the data.

Here is an example of a normal distribution:

License: CC BY-NC 3.0

[Figure1]

The line in the middle represents the mean, and the tick lines along the bottom represent the standard deviation.

Binomial Distribution

Characteristics of a Binomail Probability Distribution

  • The experiment consists of a fixed number of independent trials
  • Each trial results in one of two outcomes: success or failure
  • The probability of success p is the same for each trial
  • The binomial random variable X is defined as the number of successes in n trials

The binomial distribution is found by calculating the binomial probabilities for k = 0, 1, 2,..., n.  It can be represented in a histogram with k along the x-axis and P(k) along the y-axis.

When n * p > 10 and n(1-p) > 10, the following statements are all true:

  • the normal distribution will provide a good approximation of the binomial distribution
  • Mean for the binomial distribution: μ = np
  • Standard deviation for the binomial distribution: \sigma= √(np(1-p))

The binompdf function on the TI-83/84 calculator can be used to solve problems involving the probability of a precise number of successes out of a certain number of trials.  The syntax for the binompdf function is binompdf (n, p, a) , where n is the number of trials, p is the probability of success for any particular trial, and a is the number of successes.

The binomcdf function on the TI-83/84 calculator can be used to solve problems involving the probability of less than or equal to a number of successes out of a certain number of trials.  The syntax for the binomcdf function is binomcdf (n, p, a).

Multinomial Distribution

The multinomial distribution gives the probability of any particular combination of numbers of successes for the various categories.

This probability is given by:

P & = \frac{n!}{n_1!n_2!n_3!\ldots n_k!} \times \left (p_1{^{n_1}} \times p_2{^{n_2}} \times p_3{^{n_3}} \ldots p_k{^{n_k}} \right )

where:

n is the number of trials.

p is the probability for each possible outcome.

k is the number of possible outcomes.

Geometric Distribution

Characteristics of a Geometric Probability Distribution

  • The experiment consists of a sequence of independent trials
  • Each trial results in one of two outcomes: success or failure
  • The probability of success p is the same for each trial
  • The geometric random variable X is defined as the number of trials until the first success is observed

The geometric distribution is found by calculating the geometric probabilities for k = 0, 1, 2, ..., ∞ .  It can be represented in a histogram with k along the x-axis and P(k) along the y-axis.  Note: As k increases, P(k) approaches 0, so as this happens you can stop your calculations of P(k) for the purpose of creating the distribution.

Mean for the binomial distribution: μ = 1/p

Standard deviation for the binomial distribution: \sigma= √((1-p)/p2)

Poisson Distribution

In a binomial distribution, if the number of trials, n , gets larger and larger as the probability of success, p , gets smaller and smaller, we obtain a Poisson distribution.

Characteristics of a Poisson distribution:

  • The experiment consists of counting the number of events that will occur during a specific interval of time or in a specific distance, area, or volume.
  • The probability that an event occurs in a given time, distance, area, or volume is the same.
  • Each event is independent of all other events.

Poisson Random Variable:

The probability distributionmean, and variance of a Poisson random variable are given as follows:

p(x) &= \frac{\lambda^x e^{-\lambda}}{x!}  \quad x=0, 1, 2, 3, \ldots\\\mu &= \lambda\\\sigma^2 &= \lambda

where:

\lambda= the mean number of events in the time, distance, volume or area

e= the base of the natural logarithm

Student's T Distribution

When you use s to estimate \sigma , you must use t instead of z to complete the significance test for a mean.

In calculating the t- test statistic, we use the formula:

t=\frac{\bar{x}-\mu_0}{\frac{s}{\sqrt{n}}}

where:

t is the test statistic and has n-1 degrees of freedom.

\bar{x} is the sample mean

\mu_0 is the population mean under the null hypothesis.

s is the sample standard deviation

n is the sample size

\frac{s}{\sqrt{n}} is the estimated standard error

df = n - 1

Image Attributions

  1. [1]^ License: CC BY-NC 3.0

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