What is Random variable: Definition and 282 Discussions

In probability and statistics, a random variable, random quantity, aleatory variable, or stochastic variable is described informally as a variable whose values depend on outcomes of a random phenomenon. The formal mathematical treatment of random variables is a topic in probability theory. In that context, a random variable is understood as a measurable function defined on a probability space that maps from the sample space to the real numbers.

A random variable's possible values might represent the possible outcomes of a yet-to-be-performed experiment, or the possible outcomes of a past experiment whose already-existing value is uncertain (for example, because of imprecise measurements or quantum uncertainty). They may also conceptually represent either the results of an "objectively" random process (such as rolling a die) or the "subjective" randomness that results from incomplete knowledge of a quantity. The meaning of the probabilities assigned to the potential values of a random variable is not part of probability theory itself, but is instead related to philosophical arguments over the interpretation of probability. The mathematics works the same regardless of the particular interpretation in use.
As a function, a random variable is required to be measurable, which allows for probabilities to be assigned to sets of its potential values. It is common that the outcomes depend on some physical variables that are not predictable. For example, when tossing a fair coin, the final outcome of heads or tails depends on the uncertain physical conditions, so the outcome being observed is uncertain. The coin could get caught in a crack in the floor, but such a possibility is excluded from consideration.
The domain of a random variable is called a sample space, defined as the set of possible outcomes of a non-deterministic event. For example, in the event of a coin toss, only two possible outcomes are possible: heads or tails.
A random variable has a probability distribution, which specifies the probability of Borel subsets of its range. Random variables can be discrete, that is, taking any of a specified finite or countable list of values (having a countable range), endowed with a probability mass function that is characteristic of the random variable's probability distribution; or continuous, taking any numerical value in an interval or collection of intervals (having an uncountable range), via a probability density function that is characteristic of the random variable's probability distribution; or a mixture of both.
Two random variables with the same probability distribution can still differ in terms of their associations with, or independence from, other random variables. The realizations of a random variable, that is, the results of randomly choosing values according to the variable's probability distribution function, are called random variates.
Although the idea was originally introduced by Christiaan Huygens, the first person to think systematically in terms of random variables was Pafnuty Chebyshev.

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  1. I

    I Random variable and probability

    Hi! I'm searching for guidance and help since I don't know how to solve this problem. Here it is: a) The two-dimensional random variable (ξ,η) is uniformly distributed over the square K={(x,y): 0≤x≤1 , 0≤y≤1} . Let ζ=√ξ2+η2 me the distance between the origo and the point (ξ,η) . Calculate the...
  2. F

    MHB Understanding Uniform Random Variables: Comparing $X$ and $Y = 1-X$

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  3. K

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  4. F

    MHB Random Variable over probability space

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  5. Y

    MHB What is the correct probability for P(3<X<4|X>1)?

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  6. Y

    MHB Joint distribution of a discrete random variable

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  7. K

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  8. M

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  9. T

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  10. W

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  11. Kior

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  12. H

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  13. diracdelta

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  14. D

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  15. rayne1

    MHB Expected value of a continuous random variable

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  16. S

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  17. Jameson

    MHB Estimating variance of Poisson random variable

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  18. T

    PDF of a continuous random variable

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  19. I

    Distribution of a Function of a Random Variable

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  20. I

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  21. S

    MHB Random Variable Transformation

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  22. S

    MHB Probability function (p.f) of a random variable

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  23. D

    MHB Second moment of the Poisson random variable

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  24. B

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  25. D

    MATLAB Matlab estimate PDF from random variable X

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  26. S

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  27. D

    Complex Circle Equation with random variable attached to Z.

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  28. S

    Random variable conv. in prob. to c. How to find c?

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  29. R

    MHB Transform Random Var CDF to Standard Normal: F(x)=1-exp(-sqrt x)

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  30. P

    MHB Expected Value of Negative Binomial Random Variable

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  31. S

    Probablity: What's the p.d.f. of the random variable Z = X|X|

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  32. S

    Conditional distribution for random variable on interval

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  33. Y

    MHB Bivariate discrete random variable

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  34. D

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  35. M

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  36. I

    Moment generating function, CDF and density of a random variable

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  37. M

    How Do You Calculate P{S < t < S + R} for Independent Exponential Variables?

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  38. S

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  39. R

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  40. N

    MHB Conditional PDF of this random variable

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  41. M

    Find E[(X-mu)^k] - Normal Random Variable

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  42. J

    Define the function of density of the random variable Y.

    We selected X point from interval (-1,2). If X=x, we selected point Y from (-1,x^2). Define the function of density of the random variable Y.
  43. E

    What's the expected value of this problem (random variable)?

    Homework Statement What's the expected value of this problem (random variable)? X: represent the result of dice number 1 - result of dice number 2 example dice 1 first roll = 2; second roll = 3 dice 2 first roll = 1; second roll = 2 X = 2+3 -(1+2) = 2 what's the expected value...
  44. Jeffack

    Generate a Multivariate Random Variable

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  45. C

    Covariance Matrix of a Vector Random Variable w/ Components Related

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  46. trash

    [Probability] Expected Value of Random Variable

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  47. H

    MHB PGF of a conditional random variable

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  48. B

    MHB Proving of Y=g(X) as a continuous random variable

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  49. H

    MHB Transformation of Random Variable

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  50. R

    Showing tha a random variable is a martingale

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