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continuous random variable examples
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From Wikipedia
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 measureÎ». 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, chisquared, and others.
Intuitively, a continuous random variable is the one which can take a continuous range of values â€” 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 3Â½ 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 3Â½ 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.
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 yettobeperformed experiment, or the potential values of a quantity whose alreadyexisting 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 realvalued, 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
Realvalued 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 chisquare 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 chisquare 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 realvalued 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 nonzero 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 generallyagreedupon 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 realvalued 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
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!!!
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)
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.)
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
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