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. F

    I Random variable and elementary events

    Hello, A sample space is the set of all possible elementary events. A random "variable" is really a real-valued function that associates a single real number to every elementary events. For example, in the case of a fair die, the sample space is ##\Omega={1,2,3,4,5,6}##. Each number is an...
  2. T

    I Stopping Time in layman's words

    I have a question about intuitive meaning of stopping time in stochastics. A random variable ##\tau: \Omega \to \mathbb {N} \cup \{ \infty \}## is called a stopping time with resp to a discrete filtration ##(\mathcal F_n)_{n \in \mathbb {N}_0}##of ##\Omega ## , if for any ##n \in \mathbb{N}##...
  3. S

    Prob/Stats Probability book for basic understanding

    I want to learn topics related to combinatorics, probability theory, discrete and continuous random variables, joint pdf and cdf, limit theorems and point estimation, confidence intervals and hypothesis testing. Any recommendations for books to learn those topics? High school level or...
  4. F

    I Sample space, outcome, event, random variable, probability...

    Hello, I am solid on the following concepts but less certain on the correct understanding of what a random variable is... Random Experiment: an experiment that has an uncertain outcome. Trials: how many times we sequentially repeat a random experiment. Sample space ##S##: the set of ALL...
  5. C

    I Probability of White Ball in Box of 120 Balls: Solved!

    Problem: In a box there are ##120## balls with ## X ## of them being white and ## 120 - X ## being red for random variable ##X##. We know that ## E[ X] = 30 ##. We are taking out ## k ## balls randomly and with returning ( we return each ball we take out, so there is equal probability for each...
  6. A

    Help with random variable linear estimation

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  7. C

    I Randomly Stopped Sums vs the sum of I.I.D. Random Variables

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  8. C

    I Prove that the tail of this distribution goes to zero

    Theorem: Let ## X ## be a random variable. Then ## \lim_{s \to \infty} P( |X| \geq s ) =0 ## Proof from teacher assistant's notes: We'll show first that ## \lim_{s \to \infty} P( X \geq s ) =0 ## and ## \lim_{s \to \infty} P( X \leq -s ) =0 ##: Let ## (s_n)_{n=1}^\infty ## be a...
  9. A

    Poisson random process problem

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  10. F

    I Random variable vs Random Process

    Hello, When flipping a fair coin 4 times, the two possible outcomes for each flip are either H or T with the same probability ##P(H)=P(T)=0.5##. Why are the 4 outcomes to be considered as the realizations of 4 different random variables and not as different realizations of the same random...
  11. S

    Finding constant related to random variable

    Var (Y) = a2 . Var (X) (6.96)2 = a2 . (8.7)2 a = ± 0.8 But the answer key states that the value of a is only 0.8 Why a = -0.8 is rejected? Thanks
  12. A

    A The normal equivalent for a discrete random variable

    De normal distribution has the following form: $$\displaystyle f \left(x \right) \, = \,\frac{1}{2}~\frac{\sqrt{2}~e^{-\frac{1}{2}~\frac{\left(x -\nu \right)^{2}}{\tau ^{2}}}}{\tau ~\sqrt{\pi }}$$ and it's integral is equal to one: $$\displaystyle \int_{-\infty }^{\infty }\!1/2\,{\frac {...
  13. M

    MHB Understanding Random Variable Mapping and Probability Functions

    Hey! :giggle: What does it mean to give the mapping for a random variable? Do we have to give the outcome space and the probability function? Does it hold that $X: ( \Omega, P)\mapsto \mathbb{R}$ ? :unsure:
  14. docnet

    Continuous joint random variable

    (a) $$\int_0^1\int_0^1x+cy^2 dxdy=\int_0^1 [\frac{x^2}{2}+cxy^2]_0^1dy= \int_0^1\frac{1}{2}+cy^2 dy=[\frac{y}{2}+\frac{cy^3}{3}]_0^1=\frac{1}{2}+\frac{c}{3}=1$$ $$\Rightarrow c=\frac{3}{2}$$ (b) The marginal pdf of X is $$f_X(a)=\int_0^1 f_{X,Y}(a,b)db=\int_0^1 x+\frac{3}{2}y^2...
  15. M

    MHB Multiple choice test : random variable

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  16. M

    MHB Game : random variable for net profit

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  17. tworitdash

    A 2D space and 1D time evolution of a random field

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  18. PainterGuy

    Random variable and probability density function

    Hi, I was trying to solve the attached problem which shows its solution as well. I cannot understand how and where they are getting the equations 3.69 and 3.69A from. Are they substituting the values of θ₁ and θ₂ into Expression 1 after performing the differentiation to get equations 3.70 and...
  19. PainterGuy

    I Distribution function and random variable

    Hi, I cannot figure out how they got Table 2.1. For example, how come when x=1, F_X(x)=1/2? Could you please help me with it? Hi-resolution copy of the image: https://imagizer.imageshack.com/img923/2951/w9yTCQ.jpg
  20. A

    A Trend in an approximately exponentially distributed random variable

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  21. Armine

    Proof of a formula with two geometric random variables

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  22. U

    MHB Expectation of Conditional Expression for Exponentially Distributed RV

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  23. U

    MHB How to find PDF and Expected value of max(x,0), for a random variable x

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  24. archaic

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  25. F

    Understanding the PMF of a Random Variable: A Brief Overview

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  26. E

    B How do we interpret a random variable?

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  27. A

    A Third and fourth central moment of a random variable

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  28. F

    I Distribution of a sample random variable

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  29. WMDhamnekar

    MHB Transformation of random variable

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  30. A

    A Can we create a random variable using QED effects?

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  31. user366312

    A Question about the Poisson distributed of random variable

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  32. A

    Expected value of a function of a random variable

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  33. EEristavi

    B Random Variable - Mean and Variance

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  34. entropy1

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  35. J

    A Sum of independent random variables and Normalization

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  36. M

    I Probability function for discrete functions

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  37. M

    MHB Proving Skewness of a Random Variable $X$

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  38. P

    Poisson Random Variable probability problem

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  39. P

    Geometric Random Variable probability problem

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  40. King_Silver

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  41. Biker

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  42. mnb96

    I Equivalent definitions of random variable

    Hello, According to the Wikipedia article on random variables: If the above statement is true, then, instead of defining a (real) random variable as a function from a sample space of some probability space to the reals, could we equivalently define it as a subset of ℝ associated with a CDF?
  43. nomadreid

    I Independence of the conditions

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  44. R

    Continuous random variable transformation and marginals

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  45. I

    Normal approximation to Poisson random variable

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  46. P

    A Approximation for volatility of random variable

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  47. S

    MLE of Bivariate Vector Random Variable: Proof & Explanation

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  48. D

    MHB Question regarding a probability mass function of a random variable

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  49. D

    MHB Continuous random variable question

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  50. D

    MHB Consider the following probability mass function of a random variable x

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