What is Random variables: Definition and 350 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. X

    Conditional Probability for discrete random variables.

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  2. sunrah

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  3. J

    Transformation of Random Variables

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  4. T

    Measurability of random variables

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  5. J

    Random Variables: Calculating E[X|Y]

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  6. M

    Discrete Random Variables - Geometric Distribution

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  7. T

    Random Variables - Distribution and Expectations

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  8. C

    Exponential Random Variables and Conditional Probability Problem

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  9. J

    Sums of Independent Random Variables

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  10. S

    Proabibility - Random variables independence question

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  11. W

    What does an infinite sum of uniform random variables yield?

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  12. R

    Comparing two multivariate normal random variables

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  13. E

    Calculating CDF of Max of IID Random Variables with CDF F(x) and PDF f(x)

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  14. Y

    Some questions about random variables probability

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  15. T

    Probability - Random variables

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  16. T

    Probability - Random Variables

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  17. T

    Statisitics - Random Variables

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  18. C

    Joint pmf of 2 binomially distributed random variables

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  19. B

    MHB Sequence of normalized random variables

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  20. B

    MHB Proving the Uniform Distribution of Y from Independent Random Variables X

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  21. G

    Proving Sum of 2 Indep. Cauchy RVs is Cauchy

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  22. J

    MHB Sums of independent random variables

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  23. M

    Question about random variables

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  24. A

    Product of two uniform random variables on the interval [0,1]

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  25. E

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  26. N

    Sum of non-identical non-central Chi-square random variables.

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  27. E

    Distribution of Maximum of Two Random Variables

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  28. A

    Sum of two independent Poisson random variables

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  29. D

    Probability of sum of 5 independant random variables

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  30. A

    Transformation of Random Variables (Z = X-Y)

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  31. M

    Bus arrivals independent random variables

    Hi. Why in all literature bus arrivals are referred as independent random variables (Poisson as well)? Is there any reference where there is some math explanation except intuitive approach which of course tell that there is no correlation between 2 bus arrivals? Best regards
  32. J

    Jointly Distributed Discrete Random Variables

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  33. K

    Are W and Z equal as random variables and do they have equal expected values?

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  34. S

    Combination of two dependant discrete random variables

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  35. T

    Probability of Sum of Squares of 2 Uniform RVs < 1

    If you were to pick two random numbers on the interval [0,1], what is the probability that the sum of their squares is less than 1? That is, if you let Y_1 ~ U(0,1) and Y_2 ~ U(0,1), find P(Y_1^2 + Y^2_2 \leq 1). There is also a hint: the substitution u = 1 - y_1 may be helpful - look for a beta...
  36. M

    Continuous Random Variables

    Hi all, I was having some troubles with a practise question and thought I'd ask here. Given an r.v. X has a pdf of f(x) = k(1-x2), where -1<x<1, I found k to be 3/4. And I found the c.d.f F(x) = 3/4 * (x - x3/3 + 2/3) Now I have to find a value a such that P(-a <= X <= a) = 0.95. I thought...
  37. I

    One Function of Two Random Variables

    Homework Statement Given f(x,y) = x + y 0≤x≤1, 0≤y≤1 zero , otherwise Show that X+Y has density f(z) = z^2, 0 < z < 1, and z(2-z), 1 < z < 2. Homework Equations Also, how to find the density f(z) for X*Y? The Attempt at a Solution Even if f(z) is not given, some...
  38. M

    Sum of squared uniform random variables

    Homework Statement If X and Y are independent uniformly distributed random variables between 0 and 1, what is the probability that X^2+Y^2 is less than or equal to one. Homework Equations P(Z<1) = P(X^2+Y^2<1) For z between 0 and 1, P(X^2<z) = P(X < √z) = √z The Attempt at a Solution...
  39. ArcanaNoir

    Probability function of two random variables, another non-convergent integral

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  40. C

    How can I find the probability of X being greater than both Y and Z?

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  41. reddvoid

    Why is Y a Convolution of X1 and X2 PDFs?

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  42. M

    PMF for the sum of random variables

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  43. V

    CDF of a function of 2 random variables

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  44. G

    Convergence in probability of the sum of two random variables

    Homework Statement X, Y, (X_n)_{n>0} \text{ and } (Y_n)_{n>0} are random variables. Show that if X_n \xrightarrow{\text{P}} X and Y_n \xrightarrow{\text{P}} Y then X_n + Y_n \xrightarrow{\text{P}} X + Y Homework Equations If X_n \xrightarrow{\text{P}} X then...
  45. S

    Statistics: Proofs and Problems for Random Variables and their Distributions

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  46. R

    Distribution of Difference of 2 2nd Degree Non-Central Chi Squared RVs

    Distribution of difference of two second degree non central chi squared random variables. This problem can be cast as an indefinite quadratic form for which there are a number of general numerical techniques to determine the CDF. Alternatively, it may be written as a linear combination of...
  47. X

    Statistics question Continous Random Variables

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  48. S

    A question in random variables and random processes

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  49. C

    Density function for continuous random variables

    For the density function for random variable Y: f(y) = cy^2 for 0<= y <= 2; 0 elsewhere We are asked to find the value of c. I did a definite integral from 0 to 2 of cy^2. I get c = 3/8. Why would the book show an answer of c = 1/8? Is this an error on their part or am I missing something...
  50. S

    Conditional PDF with multiple random variables

    Homework Statement D = (L + E) / S Where L, E, and S are mutually independent random variables that are each normally distributed. I need to find (symbolically), the conditional PDF f(d|s). Homework Equations The Attempt at a Solution Not sure what to do with so many...
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