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. Jameson

    MHB Transformation of a random variable (exponential)

    Problem: Suppose that $X \text{ ~ Exp}(\lambda)$ and denote its distribution function by $F$. What is the distribution of $Y=F(X)$? My attempt: First off, I'm assuming this is asking for the CDF of $Y$. Sometimes it's not clear what terminology refers to the PDF or the CDF for me. $P[Y \le y]=...
  2. Jameson

    MHB Transformation of random variable (uniform)

    This is something that when I see the work done it makes sense, but I find it difficult to do myself. I'm also aware there is an explicit formula for doing this but that involves Jacobians and a well-defined inverse, so I think it's more intuitive to do it step-by-step. Problem: Suppose $X...
  3. A

    MHB How Does Professor Roberto's Grading System Affect Student Scores?

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  4. G

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  5. twoski

    Generating a Random Variable with a Specific Distribution Function

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  6. D

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  7. L

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  8. twoski

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  9. V

    Product of random variable with Unif dist and its variance

    First of, I apologize for the vague title, I didn't know how to summarize this issue. Homework Statement Suppose that the interest rate obtained in month i is a random variable Ri with the uniform distribution on [0.01, 0.03], where R1,R2, . . . are independent. A capital of 1 unit...
  10. twoski

    Probability and Poisson Random Variable

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  11. A

    Cdf of a discrete random variable and convergence of distributions

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  12. W

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  13. O

    MHB Discrete-continuous random variable

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  14. K

    Conditional Expectation of a random variable

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  15. C

    Q: How to compute P(X=200 and Y<150)?

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  16. S

    Continuous Functions of One Random Variable

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  17. U

    MHB Solving Random Variable Work: 0 to Infinity = 0.002?

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  18. W

    Is a random variable really random or just a function?

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  19. L

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  20. F

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  21. denjay

    Most probable value of a random variable?

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  22. E

    The distribution of function of random variable

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  23. S

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  24. W

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  25. P

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  26. J

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  27. fluidistic

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  28. C

    Finding the pdf of a random variable which is a function of another rv

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  29. lahanadar

    Random Process vs Random Variable vs Sample Space

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  30. S

    Probability: Discrete Random Variable

    Homework Statement Suppose X is a discrete random variable whose probability generating function is G(z) = z^2 * exp(4z-4) Homework Equations No idea The Attempt at a Solution I'm thinking that due to the exponent on the z term, that the exp(4z-4) would be the P[X=3] =...
  31. U

    MHB Functions of a Discrete Random Variable

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  32. P

    On conditional probability of an exponential random variable

    You are given a random exponential variable X: f(x) = λ exp(-λ x). Suppose that X = Y + Z, where Y is the integral part of X and Z is the fractional part of X: Y = IP(X), Z = FP(X). Which is the following conditional probability: P(Z < z | Y = n) for 0 ≤ z < 1 and n = 0, 1, … ?
  33. A

    Mean of a function of a random variable

    Hi, I have a random variable X with some zero-mean distribution. I have a function Y of this r.v. given by something complicated Y=(a+X)^\frac{2}{3} Is there an explicit way of finding the distribution of Y or even its mean? Thanks
  34. P

    About the definition of discrete random variable

    About the definition of "discrete random variable" Hogg and Craig stated that a discrete random variable takes on at most a finite number of values in every finite interval (“Introduction to Mathematical Statistics”, McMillan 3rd Ed, 1970, page 22). This is in contrast with the assumption that...
  35. J

    Continuous random variable (supply and demand)

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  36. K

    Limits for a truncated random variable

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  37. T

    Distribution of sum of discrete random variable

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  38. T

    Probability - Poisson Random Variable?

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  39. F

    Expectation of a Random Variable

    I know the E[X] = Integral between [-inf,inf] of X*f(x) dx Where X is normally distributed and f(x) is the PDF How do I find the expectation of X4? Bare with me because I'm useless in Latex So far what I've done is written the integral as Integral between [-inf,inf] of X4*f(x) dx...
  40. R

    Find the Mean and Variance of Random Variable Z = (5x+3)

    Homework Statement Find the Mean and Variance of Random Variable Z = (5x+3) Using data set:Using: & The Attempt at a Solution
  41. C

    Upper bound of random variable

    Dears, If a random variable is generated with the pdf of p(f) = 1/(f^x), how can I derive the upper bound or lower bound of the random variable? Thanks,
  42. C

    Upper bound of random variable

    Dears, If a random variable is generated with the pdf of p(f) = 1/(f^x), how can I derive the upper bound or lower bound of the random variable? Thanks,
  43. A

    When can I decompose a random variable $Y=X'-X''$?

    I am wondering if I can find a decomposition of Y that is absolutely continuous nto two i.i.d. random variables X' and X'' such that Y=X'-X'', where X' is also Lebesgue measure with an almost everywhere positive density w.r.t to the Lebesge mesure. My main intent is to come up with two i.i.d...
  44. Z

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  45. N

    Show that if X is a bounded random variable, then E(X) exists.

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  46. G

    Normal Distribution: PDF of a Normally Distributed Random Variable

    Homework Statement A Normally distributed random variable with mean μ has a probability density function given by _ρ_...*...((-ρ2(x-μ)2)/2δ) √2∏δ|...e^ Homework Equations Its standard deviation is given by: A)ρ2/δ B)δ/ρ C)√δ|/ρ D)ρ/√δ| E)√δ|/2ρ The Attempt at a Solution...
  47. A

    Show that a Gaussian Distribution Corresponds to a CTS random variable.

    Going over my Lecture Notes my Lecturer as Started with Show that a Gaussian Distribution Corresponds to a CTS random variable. Then she has i) Taken the f(x) = [p.d.f] and shown a) f(x) >= 0 for all x member of real numbers. b) Integral over all real numbers = 1 ii) Found the M.G.F then...
  48. T

    Probability generating function for random variable

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  49. T

    Distribution of Bernoulli random variable

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  50. M

    Determine constant c that random variable will have a t distribution?

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