Difference between revisions of "SMHS ProbabilityDistributions"
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3) Theory | 3) Theory | ||
| − | 3.1) Random variables: a random variable is a function or a mapping from a sample space into the real numbers (most of the time). In other words, a random variable assigns real values to outcomes of experiments. | + | 3.1) Random variables: a random variable is a function or a mapping from a sample space into the real numbers (most of the time). In other words, a random variable assigns real values to outcomes of experiments. |
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3.2) Probability density / mass and (cumulative) distribution functions: | 3.2) Probability density / mass and (cumulative) distribution functions: | ||
The probability density or probability mass function (pdf), for a continuous or discrete random variable, is the function defined by the probability of the subset of the sample space S, {s∈S}⊂S. p(x)=P({s∈S}|X(s)=x), all x. | The probability density or probability mass function (pdf), for a continuous or discrete random variable, is the function defined by the probability of the subset of the sample space S, {s∈S}⊂S. p(x)=P({s∈S}|X(s)=x), all x. | ||
The cumulative distribution function (cdf) F(x) of any random variable X with probability mass or density function p(x) is defined by the total probability of all {s∈S}⊂S, where X(s)≤x; F(x)=P(X≤x) | The cumulative distribution function (cdf) F(x) of any random variable X with probability mass or density function p(x) is defined by the total probability of all {s∈S}⊂S, where X(s)≤x; F(x)=P(X≤x) | ||
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| + | 3.3) Introduction to expectation and variance. | ||
| + | *Expectation: The expected value, expectation or mean, of a discrete random variable X is defined as E[X]=∑_xxP(X=x),expectation of a continuous random variable Y is defined as E[Y]=∫yP(y)dy, which is the integral over the domain of Y and P(y) is the probability density function of Y. An important property of expectation is E[aX+bY]=aE[X]+bE[Y] | ||
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| + | *Variance: The variance of a discrete random variable X is defined as VAR[X]=∑_x〖(X-E[X])^2 P(X=x)〗,variance of a continuous random variable Y is defined as VAR[Y]=∫〖(y-E[Y])^2 P(y)dy〗, which is the integral over the domain of Y and P(y) is the probability density function of Y. VAR[aX]=a^2 VAR[X]. VAR[X+Y]=VAR[X]+VAR[Y]+2COV(X,Y). | ||
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| + | *Covariance: COV(X,Y)=E[(X-E(X))(Y-E[Y])]. | ||
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| + | 3.4) Bernoulli distribution: A Bernoulli trial is an experiment whose dichotomous outcomes are random (e.g. ‘head vs. ‘tail’). | ||
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Revision as of 07:51, 10 July 2014
Scientific Methods for Health Sciences - Probability Distributions
IV. HS 850: Fundamentals
Distributions
1) Overview: Distribution is the fundamental basis of probability theory. They are two types of processes that we observe in nature – discrete and continuous distributions. The type of distribution depends on the type of data. Namely, discrete distribution is for discrete variable and continuous distribution is for continuous variable. This section aims to introduce various kinds of discrete and continuous distributions and the relationships between distributions.
- Discrete distribution: Bernoulli distribution, Binomial distribution, Multinomial distribution, Geometric distribution, Hypergeometric distribution, Negative binomial distribution, Negative multinomial distribution, Poisson distribution.
- Continuous distribution: Normal distribution, Multivariate normal distribution.
2) Motivation:
We have talked about different types of data and the fundamentals of probability theory. In order to capture and estimate the patterns of data, we introduced the concept of distribution. A probability distribution assigns a probability to each measurable subset of the possible outcomes of a random experiment. It can either be univariate or multivariate. A univariate distribution gives the probability of a single random variable while the a multivariate distribution (a joint probability distribution) gives the probability of a random vector which is a set of two or more random variables taking on various combinations of values. Consider the coin tossing experiment, what would be the distribution of the outcome?
3) Theory
3.1) Random variables: a random variable is a function or a mapping from a sample space into the real numbers (most of the time). In other words, a random variable assigns real values to outcomes of experiments.
3.2) Probability density / mass and (cumulative) distribution functions:
The probability density or probability mass function (pdf), for a continuous or discrete random variable, is the function defined by the probability of the subset of the sample space S, {s∈S}⊂S. p(x)=P({s∈S}|X(s)=x), all x.
The cumulative distribution function (cdf) F(x) of any random variable X with probability mass or density function p(x) is defined by the total probability of all {s∈S}⊂S, where X(s)≤x; F(x)=P(X≤x)
3.3) Introduction to expectation and variance.
- Expectation: The expected value, expectation or mean, of a discrete random variable X is defined as E[X]=∑_xxP(X=x),expectation of a continuous random variable Y is defined as E[Y]=∫yP(y)dy, which is the integral over the domain of Y and P(y) is the probability density function of Y. An important property of expectation is E[aX+bY]=aE[X]+bE[Y]
- Variance: The variance of a discrete random variable X is defined as VAR[X]=∑_x〖(X-E[X])^2 P(X=x)〗,variance of a continuous random variable Y is defined as VAR[Y]=∫〖(y-E[Y])^2 P(y)dy〗, which is the integral over the domain of Y and P(y) is the probability density function of Y. VAR[aX]=a^2 VAR[X]. VAR[X+Y]=VAR[X]+VAR[Y]+2COV(X,Y).
- Covariance: COV(X,Y)=E[(X-E(X))(Y-E[Y])].
3.4) Bernoulli distribution: A Bernoulli trial is an experiment whose dichotomous outcomes are random (e.g. ‘head vs. ‘tail’).
- SOCR Home page: http://www.socr.umich.edu
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