This book is both a tutorial and a textbook. This book presents an introduc- tion to probability and mathematical statistics and it is intended for students. Probability and Statistics. The Science of Uncertainty. Second Edition. Michael J. Evans and Jeffrey S. Rosenthal. University of Toronto. phenomenom is the proportion of times the outcome would occur in a very long series of repetitions. An Introduction to Basic Statistics and Probability – p. 3/

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What is statistics and what is probability? 5. 2. Discrete probability spaces. 7. 3. Examples of discrete probability spaces. 4. Countable and uncountable. Probability Theory and. Statistics. With a view towards the natural sciences. Lecture notes. Niels Richard Hansen. Department of Mathematical Sciences. PDF | This book is both a tutorial and a textbook. It is based on tion to probability and mathematical statistics and it is intended for students.

If a random variable X is given and its distribution admits a probability density function f , then the expected value of X if the expected value exists can be calculated as. Not every probability distribution has a density function: A distribution has a density function if and only if its cumulative distribution function F x is absolutely continuous.

In this case: F is almost everywhere differentiable , and its derivative can be used as probability density:. Two probability densities f and g represent the same probability distribution precisely if they differ only on a set of Lebesgue measure zero. In the field of statistical physics , a non-formal reformulation of the relation above between the derivative of the cumulative distribution function and the probability density function is generally used as the definition of the probability density function.

This alternate definition is the following:.

It is possible to represent certain discrete random variables as well as random variables involving both a continuous and a discrete part with a generalized probability density function, by using the Dirac delta function. The density of probability associated with this variable is:. More generally, if a discrete variable can take n different values among real numbers, then the associated probability density function is:. This substantially unifies the treatment of discrete and continuous probability distributions.

Probability density function

For instance, the above expression allows for determining statistical characteristics of such a discrete variable such as its mean , its variance and its kurtosis , starting from the formulas given for a continuous distribution of the probability.

It is common for probability density functions and probability mass functions to be parametrized—that is, to be characterized by unspecified parameters.


It is important to keep in mind the difference between the domain of a family of densities and the parameters of the family. Different values of the parameters describe different distributions of different random variables on the same sample space the same set of all possible values of the variable ; this sample space is the domain of the family of random variables that this family of distributions describes.

A given set of parameters describes a single distribution within the family sharing the functional form of the density. From the perspective of a given distribution, the parameters are constants, and terms in a density function that contain only parameters, but not variables, are part of the normalization factor of a distribution the multiplicative factor that ensures that the area under the density—the probability of something in the domain occurring— equals 1.

This normalization factor is outside the kernel of the distribution.

Since the parameters are constants, reparametrizing a density in terms of different parameters, to give a characterization of a different random variable in the family, means simply substituting the new parameter values into the formula in place of the old ones. Changing the domain of a probability density, however, is trickier and requires more work: For continuous random variables X 1 , This density function is defined as a function of the n variables, such that, for any domain D in the n -dimensional space of the values of the variables X 1 , This is called the marginal density function, and can be deduced from the probability density associated with the random variables X 1 , Continuous random variables X 1 , If the joint probability density function of a vector of n random variables can be factored into a product of n functions of one variable.

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This elementary example illustrates the above definition of multidimensional probability density functions in the simple case of a function of a set of two variables. If the function g is monotonic , then the resulting density function is.

This follows from the fact that the probability contained in a differential area must be invariant under change of variables. That is,.

However, rather than computing. The values of the two integrals are the same in all cases in which both X and g X actually have probability density functions.

It is not necessary that g be a one-to-one function. In some cases the latter integral is computed much more easily than the former. See Law of the unconscious statistician. The above formulas can be generalized to variables which we will again call y depending on more than one other variable. Then, the resulting density function is [ citation needed ]. This derives from the following, perhaps more intuitive representation: Suppose x is an n -dimensional random variable with joint density f.

The following formula establishes a connection between the probability density function of Y denoted by f Y y and f X i x i using the Dirac delta function:. The probability density function of the sum of two independent random variables U and V , each of which has a probability density function, is the convolution of their separate density functions:.

It is possible to generalize the previous relation to a sum of N independent random variables, with densities U 1 , Then, the joint density p y , z can be computed by a change of variables from U,V to Y,Z , and Y can be derived by marginalizing out Z from the joint density.

And the distribution of Y can be computed by marginalizing out Z:. Note that this method crucially requires that the transformation from U , V to Y , Z be bijective.

Probability and Statistics

Exactly the same method can be used to compute the distribution of other functions of multiple independent random variables. Printer-friendly version A continuous random variable takes on an uncountably infinite number of possible values.

For continuous random variables, as we shall soon see, the probability that X takes on any particular value x is 0. We'll do that using a probability density function "p. We'll first motivate a p. Example Even though a fast-food chain might advertise a hamburger as weighing a quarter-pound, you can well imagine that it is not exactly 0. One randomly selected hamburger might weigh 0.

Probability density function

What is the probability that a randomly selected hamburger weighs between 0. In reality, I'm not particularly interested in using this example just so that you'll know whether or not you've been ripped off the next time you order a hamburger! Instead, I'm interested in using the example to illustrate the idea behind a probability density function. If you weighed the hamburgers, and created a density histogram of the resulting weights, perhaps the histogram might look something like this: In this case, the histogram illustrates that most of the sampled hamburgers do indeed weigh close to 0.Two probability densities f and g represent the same probability distribution precisely if they differ only on a set of Lebesgue measure zero.

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In a more precise sense, the PDF is used to specify the probability of the random variable falling within a particular range of values , as opposed to taking on any one value.

Probability density function PDF is a statistical expression that defines a probability distribution for a continuous random variable as opposed to a discrete random variable. In the case of this example, the probability that a randomly selected hamburger weighs between 0.

Exactly the same method can be used to compute the distribution of other functions of multiple independent random variables. We'll first motivate a p.

About this book A well-balanced introduction to probability theory and mathematical statistics Featuring updated material, An Introduction to Probability and Statistics, Third Edition remains a solid overview to probability theory and mathematical statistics. Let's test this definition out on an example. Not every probability distribution has a density function: