Probability Distributions
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Questions and Answers

What is an observed distribution?

A distribution derived from data collected.

What is a standard distribution?

A distribution generated mathematically or theoretically.

Which of the following is a discrete distribution?

  • Normal distribution
  • t-distribution
  • Binomial distribution (correct)
  • None of the above

Which of the following is a continuous distribution?

<p>Normal distribution (C)</p> Signup and view all the answers

A graph of the cumulative frequency diagram is commonly known as ______.

<p>Ogives</p> Signup and view all the answers

The shape of a binomial distribution can only be symmetrical.

<p>False (B)</p> Signup and view all the answers

What are the two types of elements recognized in a binomial situation?

<p>Success and failure.</p> Signup and view all the answers

What two parameters affect the skewness of the binomial distribution?

<p>Sample size (n) and the population proportion of elements of the first type (p)</p> Signup and view all the answers

What does a Poisson distribution describe?

<p>The occurrence of isolated events within a continuum.</p> Signup and view all the answers

A poisson distribution, is derived from the binomial with a finite sample size

<p>False (B)</p> Signup and view all the answers

Under what conditions is the Poisson distribution a good approximation of the binomial?

<p>When n is greater than or equal to 20, and p is less than or equal to 0.05.</p> Signup and view all the answers

Flashcards

Observed Distribution

A distribution derived from actual collected data.

Standard Distribution

A distribution generated mathematically or theoretically.

Discrete Distribution

A type of distribution with isolated whole number values.

Continuous Distribution

A distribution with values that can take any number in an interval.

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Frequency Table

A table that displays the frequency of different outcomes.

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Probability Histogram

A histogram that shows the probability of different outcomes.

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Ogive

A graph that represents cumulative frequencies.

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Binomial Distribution

A discrete distribution representing the probability of two outcomes in independent trials.

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Binomial Probability Formula

P(r) = nCr * p^r * (1-p)^(n-r).

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Parameters of Binomial

Defined by the number of trials (n) and the probability of success (p).

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Cumulative Probability

The sum of probabilities for all outcomes up to a certain point.

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Poisson Distribution

A discrete distribution for rare events happening within a fixed interval.

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Poisson Probability Formula

P(r events) = (e^(-λ) * λ^r) / r!

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Lambda (λ)

The average number of events in a given interval for the Poisson distribution.

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Skewness of Distribution

The asymmetry of the distribution shape based on parameters.

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Normal Distribution

A continuous distribution that is symmetrical around the mean.

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t-Distribution

A type of continuous probability distribution used with smaller sample sizes.

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Probability of Success

The likelihood of a favorable outcome in an experiment.

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Event Occurrence

The happening of a specific outcome in a probability scenario.

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Independent Trials

Experiments where the outcome of one trial does not affect another.

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Combination (nCr)

The selection of r items from n without regard to order.

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Sample Size (n)

The number of observations or trials in an experiment.

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Event Non-occurrence

When a specific event does not happen.

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Empirical Data

Data gathered through observation or experimentation.

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Probability Calculation

Finding the likelihood of an event based on known data.

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More than X

The probability of events exceeding a certain number.

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Less than X

The probability of events less than a certain number.

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Approximation of Distributions

Using one distribution to estimate another.

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Cumulative Probability Table

A reference table for quickly finding cumulative probabilities.

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Statistical Inference

The process of drawing conclusions about populations based on sample data.

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Study Notes

Probability Distributions in Biology

  • Observed Distributions are derived from collected data, unique to a specific situation. Data is measured for a variable, which changes with observation. Data is sorted into an ordered array, categorized, and shown using frequency tables and histograms for clearer visualisation (descriptive statistics). A probability histogram can also be used when analysis is required. Probability for a class can be determined by dividing the frequency of that class by the total frequency. Cumulative frequency diagrams are often graphed as ogives (less than and more than ogives).

Standard Distributions

  • Standard distributions are mathematically derived or theoretically calculated models. These distributions are classified into discrete and continuous distributions.

Discrete Distributions

  • Discrete distributions have values that are numerical whole numbers, isolated points on the number line.
  • Examples include the binomial distribution and the Poisson distribution

Continuous Distributions

  • Continuous distributions have values that are a range of continuous numbers on a number line.
  • Examples include the normal distribution and the t-distribution

Binomial Distribution

  • Definition: This distribution describes the probability of a certain number of successes in a fixed number of independent trials.

  • Characteristics:

    • Each trial has two possible outcomes: success or failure.
    • The probability of success (p) remains constant for each trial.
    • The trials are independent.
    • The number of trials (n) is fixed.

  • Formula

    • P(r of Type 1 in sample) = "nCr.pr.(1-p)n-r
    • n = sample size
    • p = proportion of Type 1 in population
    • (1-p) = proportion of Type 2 in population
    • nCr = combination

  • Shape: Skewed right, skewed left, or symmetrical, depending on the situation. This depends on both n and p values.

Poisson Distribution

  • Definition: This distribution models the probability of a given number of events occurring in a fixed interval of time or space, if these events occur with a known average rate and independently of the time since the last event.
  • Characteristics:
    • Events occur randomly in time and space.
    • The average rate of occurrence is constant.
    • The occurrence of one event does not affect the probability that another event will occur.
  • Formula:
    • P(r events) = e-λ .λr / r!
    • λ (lambda): average number of events per sample
    • e: mathematical constant approximately equal to 2.718
  • Shape: Right-skewed to almost symmetrical, depending on the average rate.

Poisson as Approximation to Binomial

  • Useful when binomial table and formula is too tedious.
  • Poisson approximates binomial when:
  • n (sample size) is greater than or equal to 20
  • p (probability of success) is less than or equal to .05
  • Then λ (lambda) = n x p

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Related Documents

LU2 Probability Distributions PDF

Description

The content covers observed and standard probability distributions in biology. Observed distributions are derived from collected data, while standard distributions are mathematically derived models. Discrete distributions, such as the binomial distribution, are also discussed.

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