#### Stochastic Processes

A stochastic or random process as opposed to a deterministic process, includes the possibility that a system may evolve in different ways in a way that can only be predicted with probability. For example a differential equation can be solved and the...

https://astarmathsandphysics.com/university-maths-notes/probability-and-statistics/2032-stochastic-processes.html#### The Beta Distribution

A random variable is said to have a beta distribution with parameters if the pdf of is The case gives the standard beta distributions. Some are shown below. Changing and only shifts and scales the graphs. Unless and are integers integration is...

https://astarmathsandphysics.com/university-maths-notes/probability-and-statistics/2033-the-beta-distribution.html#### The Bivariate Normal Distribution

The bivariate normal distribution for two related, normally distributed variables and is where If and are independent then and The bivariate normal PDF defines a surface in the plane. Like its one dimensional counterpart, the bivariate normal...

https://astarmathsandphysics.com/university-maths-notes/probability-and-statistics/2034-the-bivariate-normal-distribution.html#### The F - Distribution

The - distribution arises as the ratio of two chi-squared distributions: where and have chi-square distributions with and degrees of freedom respectively, and and are independent. The mean, variance and skew are and respectively. The probability...

https://astarmathsandphysics.com/university-maths-notes/probability-and-statistics/2035-the-f-distribution.html#### The Gamma Distribution

The gamma distribution is a continuous positively skewed probability distribution with two parameters where The standard gamma distribution has The exponential distribution is a special case of the gamma distribution with The mean is given by and the...

https://astarmathsandphysics.com/university-maths-notes/probability-and-statistics/2036-the-gamma-distribution.html#### The Hypergeometric Distribution

The hypergeometric distribution is closely related to the binomial distribution. The binomial distribution is the model for sampling with replacement from a finite collection, or sampling with or without replacement from an infinite collection, with...

https://astarmathsandphysics.com/university-maths-notes/probability-and-statistics/2037-the-hypergeometric-distribution.html#### The Lognormal Distribution

A non – negative random variable is said to have a lognormal distribution if the random variable has a normal distribution. The resulting probability density function of a lognormal random variable when is normally distributed with parameters and is It...

https://astarmathsandphysics.com/university-maths-notes/probability-and-statistics/2038-the-lognormal-distribution.html#### The Mann - Whitney - Wilcoxon Test

The Mann – Whitney – Wilcoxon test may be used to test the hypothesis that two independent samples A and B arise from the same population. The test is very nearly as powerful as the two sample t – test, and may be used where the t – test may not...

https://astarmathsandphysics.com/university-maths-notes/probability-and-statistics/2039-the-mann-whitney-wilcoxon-test.html#### The Method of Maximum Likelihood

Suppose we are trying to measure the true mean of some quantity. We make repeated measurements Intuitively we say the true value of the mean is likely to be close to the mean of our measurements, The maximum likelihood method is a general method for...

https://astarmathsandphysics.com/university-maths-notes/probability-and-statistics/2040-the-method-of-maximum-likelihood.html#### The Negative Binomial Distribution

The negative binomial distribution has the following requirements: A sequence of independent trials. Each trial can result in success (S) or failure (F). The probability of success, is a constant. Trials continue until r successes have been obtained....

https://astarmathsandphysics.com/university-maths-notes/probability-and-statistics/2041-the-negative-binomial-distribution.html#### The Pareto Distribution

The probability density function of a Pareto distribution is This is shown below for and various values of The cumulative distribution function is then found by integration: for and 0 otherwise. The mean and variance are for and for respectively....

https://astarmathsandphysics.com/university-maths-notes/probability-and-statistics/2042-the-pareto-distribution.html#### The Triangular Distribution

The triangular distribution is a continuous probability distribution with probability density function Where are a and b are the upper and lower limits of the distribution. The probability distribution function is shown below. The cumulative...

https://astarmathsandphysics.com/university-maths-notes/probability-and-statistics/2043-the-triangular-distribution.html#### The Two Sample t - Test

The two sample t – test is one of the most useful and widely used statistical tests. It tests for the equality of the means of two samples, subject to the assumptions: The two sample both arise from normal distributions. The variances of the...

https://astarmathsandphysics.com/university-maths-notes/probability-and-statistics/2044-the-two-sample-t-test.html#### The Weibull Distribution

The probability density function of a Weibull random variable X is where is the shape parameter and is the scale parameter. This is shown below. If is a "time-to-failure", the distribution gives a distribution for which the failure rate is...

https://astarmathsandphysics.com/university-maths-notes/probability-and-statistics/2045-the-weibull-distribution.html#### The Wilcoxon Signed Rank Test

The Wilcoxon signed rank test assumes only a continuous and symmetric distribution with mean =median = If we have a sample then we find and rank them from smallest to largest. The null hypothesis is The test statistic is the sum of the ranks of those...

https://astarmathsandphysics.com/university-maths-notes/probability-and-statistics/2046-the-wilcoxon-signed-rank-test.html#### Derivation of Expressions for Multiple Regression Coefficients

Supose we suspect that an observable \[Y\] and random variables \[X+_i , \: i=1,2,3,...n\] are linearly related. We can try to find a relationship of the form \[y =\beta_0+\beta_1(x_{1}- \bar{x}_1)+ \beta_2(x_{2}-\bar{x}_2))+...+\beta_n(x_{n}-...

https://astarmathsandphysics.com/university-maths-notes/probability-and-statistics/4708-derivation-of-expressions-for-multiple-regression-coefficients.html#### Fitting Data Points to a Quadric Expression Using Least Squares Method

We can model a set of data points to a quadratic function using the least squares method. Suppose we have a set of points \[\{(x_1,y_i)@i=1,2,...,n \}\] and we want to find a quadratic expression \[y=a+bx+cx^2\] to model this data. The least squares...

https://astarmathsandphysics.com/university-maths-notes/probability-and-statistics/4802-fitting-data-points-to-a-quadric-expression-using-least-squares-method.html#### Lest Sequare Fit to an Exponential Relationship

Suppose we have variables \[x, \: y\] exponentially related, so that y=Ae^{Bx} \[\] , and we have data points \[\{(x_1,y_1),(x_2,y_2),....,(x_n,y_n) \}\] from which we wish to find estimates for \[A, \: B\] . We can estimate \[ln(A), \: B\] by...

https://astarmathsandphysics.com/university-maths-notes/probability-and-statistics/4804-lest-sequare-fit-to-an-exponential-relationship.html#### Linear Regression Line on Two Independent Variables Example

Suppose we want to find the regression line of \[y\] on independent variabbles \[x_1 \: x_2, \:,..., x_k\] from the set of data points \[\{ (x_{i1j},x_{i2},...,x_{nk} ,y_i): i=1,2,...,n\}\] . The regression line will take the form \[y=\beta_0 +...

https://astarmathsandphysics.com/university-maths-notes/probability-and-statistics/4805-linear-regression-line-on-two-independent-variables-example.html#### Advanced Questions - Venn Diagrams

More complicated question on Venn diagrams involve the use of simultaneous equations, or the introduction of a dummy variable x. Example Sketch the Venn diagram. (1) (2) Divide (1) by (2) to get Substitute these into the equation Then and from (1) We...

https://astarmathsandphysics.com/ib-maths-notes/probability-and-statistics/1039-advanced-questions-on-venn-diagrams.html