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From Wikipedia

Continuous probability distribution

In probability theory, a continuous probability distribution is a probability distribution which possesses a probability density function. Mathematicians also call such distribution absolutely continuous, since its cumulative distribution function is absolutely continuous with respect to the Lebesgue measureλ. If the distribution of X is continuous, then X is called a continuous random variable. There are many examples of continuous probability distributions: normal, uniform, chi-squared, and others.

Intuitively, a continuous random variable is the one which can take a continuous range of values — as opposed to a discrete distribution, where the set of possible values for the random variable is at most countable. While for a discrete distribution an event with probability zero is impossible (e.g. rolling 3½ on a standard die is impossible, and has probability zero), this is not so in the case of a continuous random variable. For example, if one measures the width of an oak leaf, the result of 3½ cm is possible, however it has probability zero because there are infinitely many other potential values even between 3 cm and 4 cm. Each of these individual outcomes has probability zero, yet the probability that the outcome will fall into the interval is nonzero. This apparent paradox is resolved by the fact that the probability that X attains some value within an infinite set, such as an interval, cannot be found by naively adding the probabilities for individual values. Formally, each value has an infinitesimally small probability, which statistically is equivalent to zero.

Formally, if X is a continuous random variable, then it has a probability density functionÆ’(x), and therefore its probability to fall into a given interval, say is given by the integral

\Pr[a\le X\le b] = \int_a^b f(x) \, dx In particular, the probability for X to take any single value a (that is ) is zero, because an integral with coinciding upper and lower limits is always equal to zero.

The definition states that a continuous probability distribution must possess a density, or equivalently, its cumulative distribution function be absolutely continuous. This requirement is stronger than simple continuity of the cdf, and there is a special class of distributions, singular distributions, which are neither continuous nor discrete nor their mixture. An example is given by theCantor distribution. Such singular distributions however are never encountered in practice.

Note on terminology: some authors use the term to denote the distribution with continuous cdf. Thus, their definition includes both the (absolutely) continuous and singular distributions.


Random variable

In probability and statistics, a random variable or stochastic variable is a variable whose value is not known. Its possible values might represent the possible outcomes of a yet-to-be-performed experiment, or the potential values of a quantity whose already-existing value is uncertain (e.g., as a result of incomplete information or imprecise measurements). Intuitively, a random variable can be thought of as a quantity whose value is not fixed, but which can take on different values; a probability distribution is used to describe the probabilities of different values occurring. Realizations of a random variable are called random variates.

Random variables are usually real-valued, but one can consider arbitrary types such as boolean values, complex numbers, vectors, matrices, sequences, trees, sets, shapes, manifolds, functions, and processes. The term random elementis used to encompass all such related concepts. A related concept is thestochastic process, a set of indexed random variables (typically indexed by time or space).

Introduction

Real-valued random variables (those whose range is the real numbers) are used in the sciences to make predictions based on data obtained from scientific experiments. In addition to scientific applications, random variables were developed for the analysis of games of chance and stochastic events. In such instances, the function that maps the outcome to a real number is often the identity function or similarly trivial function, and not explicitly described. In many cases, however, it is useful to consider random variables that are functions of other random variables, and then the mapping function included in the definition of a random variable becomes important. As an example, the square of a random variable distributed according to a standard normal distribution is itself a random variable, with a chi-square distribution. One way to think of this is to imagine generating a large number of samples from a standard normal distribution, squaring each one, and plotting a histogram of the values observed. With enough samples, the graph of the histogram will approximate the density function of a chi-square distribution with one degree of freedom.

Another example is the sample mean, which is the average of a number of samples. When these samples are independent observations of the same random event they can be called independent identically distributed random variables. Since each sample is a random variable, the sample mean is a function of random variables and hence a random variable itself, whose distribution can be computed and properties determined.

One of the reasons that real-valued random variables are so commonly considered is that the expected value (a type of average) and variance (a measure of the "spread", or extent to which the values are dispersed) of the variable can be computed.

There are two types of random variables: discrete and continuous. A discrete random variable maps outcomes to values of a countable set (e.g., the integers), with each value in the range having probability greater than or equal to zero. A continuous random variable maps outcomes to values of an uncountable set (e.g., the real numbers). For a continuous random variable, the probability of any specific value is zero, whereas the probability of some infinite set of values (such as an interval of non-zero length) may be positive. A random variable can be "mixed", with part of its probability spread out over an interval like a typical continuous variable, and part of it concentrated on particular values like a discrete variable. These classifications are equivalent to the categorization of probability distributions.

The expected value of random vectors, random matrices, and similar aggregates of fixed structure is defined as the aggregation of the expected value computed over each individual element. The concept of "variance of a random vector" is normally expressed through a covariance matrix. No generally-agreed-upon definition of expected value or variance exists for cases other than just discussed.

Examples

The possible outcomes for one coin toss can be described by the state space \Omega = {heads, tails}. We can introduce a real-valued random variable Y as follows:

Y(\omega) = \begin{cases} 1, & \text{if} \ \ \omega = \text{heads} ,\\ 0, & \text{if} \ \ \omega = \text{tails} . \end{cases}

If the coin is equally likely to land on either side then it has a probability mass function given by:

\rho_Y(y) = \begin{cases}\frac{1}{2},& \text{if }y=1,\\

\frac{1}{2},& \text{if }y=0.\end{cases}


From Yahoo Answers

Question:Two dice are rolled and the two numbers on the uppermost faces are added together. What is the probability that this total is (a) six, (b) more than six (c) this dice rolling is carried out 10 times. What is the probability that a total of more than six will occur on exactly 4 occasions. I'm up to learning how to approximate the binomial distribution to a normal distribution, and this question just really confuses me. It's very different from the examples in my book and every time I worked out the answer, it's way off from the real actual answer. My answer for part (a) is 1/6, but apparently it's wrong. I have no idea what i need to do for these questions, so could someone please explain it to me? Thanks Stanschim: Erm..is there a quick and easier way to do part (a)?

Answers:Lets set up the distribution; there are 36 possible outcomes, each equally likely. Die1 Die2 Total 1 1 2 1 2 3 1 3 4 1 4 5 1 5 6 1 6 7 2 1 3 2 2 4 2 3 5 2 4 6 2 5 7 2 6 8 3 1 4 3 2 5 3 3 6 3 4 7 3 5 8 3 6 9 4 1 5 4 2 6 4 3 7 4 4 8 4 5 9 4 6 10 5 1 6 5 2 7 5 3 8 5 4 9 5 5 10 5 6 11 6 1 7 6 2 8 6 3 9 6 4 10 6 5 11 6 6 12 a) There are 5 ways to get 6, so the probability is 5/36. b) There are 21 ways to get more than 6, so the probability is 21/36 = 7/12. c) We just showed that the probability of getting more than 6 is 7/12. We can use that with the binomial distribution to answer your question. (10 nCr 4)(7/12)^(4)(5/12)^6 = 0.1272386405 There is no substitute for "long and boring." Probability is about counting possible events (the denominator) and counting the desired outcomes (the numerator). So, I would say there is no shortcut - boring doesn't take as long as you think!!!

Question:A) # of petals on a randomly chosen daisy. B) Stem length in cm of a random daisy C) # of daisys found in a randomly chosen grassy area 1 sq meter in size. D) average number of petals per daisy computed from all the daisies found in a randomly chosen grassy area 1 sq meter in size. I think A) is discrete and B) is continuous, but what about the rest and why> Thank you.

Answers:you were correct with A and B. C is discrete ( you can't have a fraction of a daisy - only a whole number) D is continuous ( the average could be any number)

Question:Please help..I trying to help my grandson You just can give the steps A continuous random variable X that can assume values only between X=2 and X=8 inclusive has a density function given by f(x) = 1/48(x+3) Show that it is complete p.d.f also Find E(X).

Answers:You must show that the integral of the density function is 1 The integral of 1/48(x + 3) = 1/48 (1/2 x^2 + 3x) Evaluated from 2 to 8 = 1/48 ((1/2(8^2) + 3 (8)) - (1/2(2^2) + 3(2))) = 1 E(x) is the integral of x(f(x))dx = 1/48 (1/3 (x^3) + 1/2 (x^2)) Evaluated from 2 to 8 1/48 ((1/3(8^3) + 1/2(8^2)) - (1/3(2^3) + 1/2(2^2))) = 99/24 (double check, I'm working by hand.)

Question:Is a discrete random variable difined by a probability density function? And is as continuous random variable defined by just a probability distribution? Thanks Vikram!

Answers:A discrete random variable is one which may take on only a countable number of distinct values such as 0,1,2,3,4,........ Discrete random variables are usually (but not necessarily) counts. If a random variable can take only a finite number of distinct values, then it must be discrete. Examples of discrete random variables include the number of children in a family, the Friday night attendance at a cinema, the number of patients in a doctor's surgery, the number of defective light bulbs in a box of ten. A continuous random variable is one which takes an infinite number of possible values. Continuous random variables are usually measurements. Examples include height, weight, the amount of sugar in an orange, the time required to run a mile. If you want more detailed answer then visit http://en.wikipedia.org/wiki/Random_variable

From Youtube

Example on finding the median and quartiles of a continuous random variable. :In this video I run through a numerical example on finding the median and lower and upper quartiles of a continuous random variable from its probability density function. To see this video in a larger clearer form goto ExamSolutions at www.examsolutions.co.uk

Example on finding the Mean E(X) and Variance Var(X) for a Continuous Random Variable :In this example you are shown how to calculate the mean, E(X) and the variance Var(X) for a continuous random variable. To see this video in a larger clearer form goto ExamSolutions at www.examsolutions.co.uk