Tutorial

1

Cauchy as the distribution of tg(q).
   Cauchy as the ratio of two standard normal r.v..
   Reciprocal of Cauchy is Cauchy.
   Sample mean of the Cauchy distribution is Cauchy.

Animation

The distribution of the sample mean does not depend on the     sample size, and is therefore identical to the "mother" distribution.

Tutorial

1

Basic properties of the Chi-square distribution.

2

Distribution of the sample variance.
   Loss a degree of freedom.
   Independence of the sample mean and the sample variance.

Animation

Shapes of the Chi-square distribution (number of degrees of     freedom adjustbale).
    Distribution of the true or estimated variance.

Tutorials

1

Basic properties of the exponential distribution.

2

Estimating the mean by the method of Maximum Likelihood.

3

Memoryless property of the exponential distribution.

4

Strong memoryless property of the exponential distribution.

5

Hazard rate function, failure rate.

6

Probability for a component to be the first one to fail.

7

Components in series, parallel, and on standby.

8

Expected lifetime of a system made of components in parallel.    Distributions of the spacings of the exponential distribution.

9

The "2// vs. 1" problem.

10

Random (geometric) sum of exponential variables.

11

Distributions of X₁ and of X₂ under the condition X₁ < X₂

Animations

Basic properties (mean, mode, standard deviation).

Estimating the mean by the method of Maximum Likelihood.

Memoryless property of the exponential distribution (probabilistic     viewpoint).

Memoryless property of the exponential distribution (distributional     viewpoint).
   Strong memoryless property of the exponential distribution.

Probability for a component to be the first one to fail.

Distribution of the min of several independent exponentials.

Expected lifetime of an "all parallel" system.
   Distribution of the spacings of the exponential distribution.

Distributions of X₁ and of X₂ under the condition X₁ < X₂

Tutorials

1

Probability density function of Fisher's F distribution.
   Mode.
   Mean, second order moment, variance.
   Distribution of the inverse of a Fisher random variable.

2

Mean and variance : second method.

See also :
       * Transforming a F distribution into a Beta distribution.

Animation

Shapes of Fisher's distribution depending of the values of the    numbers of degrees of freedom.

Tutorial

1

Recurrence relation of the Gamma function.
   Moments of all orders, variance.
   Moment generating function.
   Additivity. Sum of i.i.d. exponential r.v..
   Cumulative distribution function when α is an integer.
   Relationship with the Poisson distribution.

Animations

Shapes of the Gamma distribution as a function of the values of
   parameters α and β.

The Gamma distribution as a sum of iid exponentials.

Tutorials

1

Basic properties of the normal distribution

2

Estimation of the parameters of the normal distribution

3

Basic properties of normally distributed random variables

4

Normal distribution whose mean is itself a
   normally distributed r.v..

Animations

Sample mean is normally distributed

Estimation of the parameters of the normal distribution
   by the method of Maximum Likelihood

Normal distribution whose mean is itself a
   normally distributed r.v..

Tutorials

1

The standard bivariate normal distribution.
   The general bivariate normal distribution.

2

Uncorrelated yet not independent normal r.v..

Animations

Binormal distribution. Influence of :
        * The standard deviations,
        * and the correlation coefficient
    of the marginal distributions.

Two examples of uncorrelated yet not independent normal r.v..

Tutorials

1

Distributions of the order statistics of the uniform distribution.
   Distributions of the order statistics of any distribution. 

2

Joint distribution of all order statistics.
   Joint distribution of two order statistics.

Animation

Distributions of the order statistics of the uniform distribution.

Tutorials

1

Probability density function of Student's t distribution.
   Special cases : Cauchy, Normal (n infinite).
   Welch's approximation.

2

Variance of Student's t distribution :
       1) Direct approach by Wallis integrals.
       2) By LOTUS Theorem.

Animation

Student's t, normal and Cauchy distributions.
   Progressive histogram of Student's t distribution.

Tutorials

1

Basic properties of the uniform distribution

2

Some applications of the uniform distribution.

3

Solutions to the four problems in the Glossary.

Animation

Four illustrated problems :
       * The "broken stick" distribution.
       * The "Shortest segment" problem.
       * The "Longest segment" problem.
       * The "Random choice" problem.

Tutorial

1

Bivariate transformations form first principles.