Random variable Definition and 282 Threads

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. R

    How is the PDF of a Non One-to-One Transformation Derived for n Dimensions?

    This is not homework. Case I is mostly for background. The real questions are in Case II. Case I (one dimension): a. Suppose X is a continuous r.v. with pdf fX(x), y = g(x) is one-to-one, and the inverse x = g-1(y) exists. Then the pdf of Y = g(X) is found by f_Y(y) = f_X(g^{-1}(y) |...
  2. M

    Expectation of 2 random variable, E(|X-Y|^a)

    Hi, anyone help please. Let X and Y are independent uniform random variables over the interval [0,1] E[|X-Y|a]=? where, a>0
  3. M

    Subtracting out a random variable

    "Subtracting out" a random variable let X be a discrete R.V. and let Y = f(X) for some function f. I wish to find a function g, such that Y and Z = g(X) are independent, and also such that the uncertainty H(Z) is maximized. For example, suppose X is uniformly distributed over...
  4. E

    Distribution of Exponential Random Variable (RV)

    Hello, In a paper, the authors defined an exponential Random Variable (RV) as X_1 \mbox{~EXP}(\lambda) where \lambda is the hazard rate. What will be the distribution of this RV: f_{X_1}(x)=\lambda e^{-\lambda x} or f_{X_1}(x)=\frac{1}{\lambda} e^{-\frac{x}{\lambda}} Thanks in advance.
  5. A

    How Do You Calculate Probabilities for Normally Distributed Scores?

    Continous Random Variable HELP PLEASE! Scores on a particular test are normally distributed in the population, with a mean of 100 and a standard deviation of 15. What percentage of the population have scores ... a) Between 100 and 125 b) Between 82 and 106 c) Between 110 and 132...
  6. A

    Cauchy Random Variable Homework with Equations and Attempt

    Homework Statement I have attached the problem statement Homework Equations Also find attached The Attempt at a Solution My attempt is attached together with the problem statement and the relevant equations.
  7. L

    Probability Density Function with an exponential random variable

    The question is: if X is an exponential random variable with parameter \lambda = 1, compute the probability density function of the random variable Y defined by Y = \log X. I did F_Y(y) = P \{ Y \leq y \} = P \{\log X \leq y \} = P \{ X \leq e^y \} = \int_{0}^{e^y} \lambda e^{- \lambda x} dx =...
  8. B

    What is the Density Function of Y=√X?

    Homework Statement Suppose a random variable X has probability density function(pdf) f(x) { 1/3 for 1 \leq x \leq 4 find the density function of Y= \sqrt{X} The Attempt at a Solution y=g(x)=\sqrt{x} so g^-1(y)=x=y^2 A= \{ x: 1 \leq x \leq 4 \} is monotonic onto B= \{y: 1 \leq...
  9. M

    Estimating & Plotting PDF & CDF of Random Variable X in MATLAB

    halw could anyone help me in writing this project in MATLAB ?? A random variable X is observed at certain experiment. 100,000 samples of this random variable are stored in a vector called samples. 1. Use MATLAB to read the samples of this random variable. To read these samples you...
  10. A

    Distribution of Log of Random Variable

    Let X and Y be random variables. X ~ N(u,s^2) Y = r ln X, where r is a constant. What is the distribution of Y? (This is not a homework problem. It's just related to something I was curious about, and I can't figure out how to solve this, if it is solvable...)
  11. S

    Probability of a random variable

    Homework Statement 5 men and 5 women are ranked according to exam scores. Assume no two scores are the same and each 10! rankings are equally likely. Let random variable X denote the highest ranking achieved by a woman e.g. X=2 means the highest test score was achieved by 1 of the 5 men and the...
  12. P

    Continuous Random Variable question

    Homework Statement Problem statement is underlined. Having problems to prove this. Homework Equations F(x) = ∫ f(x) dx Question relating to cumulative distributive function. Part ii requiring to relate cumulative distributive function to probability density function. The Attempt...
  13. U

    E(X): Find Probability of Rolling 4 Consecutive 6's with a Fair Dice

    Let X be a random variable representing the number of times you need to roll (including the last roll) a fair six-sided dice until you get 4 consecutive 6's. Find E(X)? answer is 1554. I get confused with this, probability { X > n-5 }. I know that the last for throws must be 6's and the one...
  14. S

    Find cdf, pdf and expextation value of a random variable

    Homework Statement Let X represent the random choice of a real number on the interval [-1,1] which has a uniform distribution such that the probability density function isf_{X}(x)=\frac{1}{2} when -1\leqx\leq1. Let Y=X^{2} a. Find the cumulative distribution F_{Y}(y) b. the density function...
  15. J

    Constant times a random variable question

    A random variable X follows a certain distribution. Now say I multiply the random variable X by a constant a. Does the new random variable aX follow the same distribution as X?
  16. C

    Compute Variance of Random Variable X

    Homework Statement Compute the variance of the random variable X given by V(X) = \sqrt{E((X-E(X))^2)} where E(X) is the expectation value of random variable X Homework Equations Hint: Use parameter differentiation The Attempt at a Solution I have no idea what to do here. I've never taken...
  17. B

    How to Calculate E(XY) for Random Variables X and Y?

    I have two random variables X and Y, and I need to calculate E(XY). The expectation of X, E(X) = aZ, and the expectation of Y, E(Y) = bZ, where a and b are known constants and Z is a random variable. So the question is how would I calculate E(XY)? I was thinking that I could do the...
  18. X

    Resistance as a Random Variable?

    I've been asked to model a resistance as a random variable. I am not exactly sure what that entails, but I was hoping someone might give me a little bit of insight. I have one resistance that is oscillating sinusoidally, and that produces a U shaped probability distribution. As is shown in...
  19. N

    Suppose x is a discrete, binomial random variable

    How do I do this p(x<1) this sign has a _ under the < n=6 p=0.1 Suppose x is a discrete, binomial random variable. Calculate P(x > 2), given trails n = 8, success probability p = 0.3 [Hint: P(x > value) = 1 – P(x <= value) <= is a < with a _ under it (tell me the number...
  20. L

    Expectation of random variable is constant?

    hi there. currently looking at the two conditions that must be met for a process to be wide sense stationary. The first constion is: E[X(t)] = constant what exactly does this mean??isn't is obvious that any random variable (with fixed time) will always yield a constant expextation. I...
  21. S

    Regarding continuous random variable

    Here's the qn random variable X follows uniform distribution [-a,a] and random variable Y is defined as Y=e^x find E(Y) i figure that E(Y)=E(e^x) but somehow can't carry on from there can anyone help?
  22. P

    Generating a normal random variable

    I'm trying to write a program in excel to generate random variables with mean mu and standard deviation sigma. I can simply refer to the worksheet function for it but it takes forever when I have it inside a loop doing a monte carlo simulation. There is one function in excel that returns a...
  23. J

    Is the Absolute Value of a Random Variable Also a Random Variable?

    I have a question, we know that a random variable X is a function that maps a real number to an event in the Sample Space. But, if X is a random variable, then the absolute value of X, say |X| is a random variable too? Why? I am almost sure that it is not because we can not tell wheter an...
  24. -Job-

    Binomial Random Variable With Non-Integer value

    So the problem gives a binomial random variable X with parameters n=5 and p=0.25 and ask for the probability P(X=1.5). The binomial probability mass function is defined only for integers. Should i approximate using the normal distribution or the poisson?
  25. R

    Expectation value of continuous random variable

    Hi.. i am doing this question for Probability Theory, to find E[x] of a continuous random variable E[x] = the integral from (0 to infinity) of 2x^2 * e^(-x^2) dx So I used integration by parts... u = x^2 du = 2xdx dv = e^(-x^2) <--- ahh... how do you integrate that. (it dosn't look like...
  26. P

    Geometric random variable with probability mass function

    If you have a geometric random variable with probability mass function: P(X=n) = p(1-p)^n n = 0,1,2,3... Find the Mean and the Variance. ---------------------------- Okay, I've looked everywhere and tried everything, however, i just cannot get it. i think that your supposed...
  27. P

    Calculating Expected Value using Probability Mass Function for Random Variable X

    Hello... hmm.. i am working on a homework problem, and I am kina stuck. the question reads: Suppose that X is a random variable which can take on any non-negative integer (including 0). Write P(X greater than and equal to i) in terms of the probability mass function of X and hence show...
  28. D

    Continuous Random variable Problem

    Hi,I'm new to the site. I come with a question I was hoping soemone out there can help me set up. "The manager of a bakery knows that the number of chocalate cakes he can sell on any given day is a random variable with the probability mass function: p(x)=1/6 For x=0,1,2,3,4,5...
  29. S

    Conditional Prob -cont random variable

    hello all I have been workin on some problems involving conditional probability and continuous random variables and the thing is i don't know if i get the limits correct, anyway here is the problem, check it out, any suggestions would be helpful f(y_1,y_2)...
  30. N

    THe Laplace random variable has a PDF that is a double exponential

    THe Laplace random variable has a PDF that is a double exponential, fT(t)=ae^(-|t|/2) for all values of t and a, a constant to be determined. A) Find a (Answer 1/4) B)Find the expected value of T, given T is greater than or equal to -1. (Answer 1.31)
  31. F

    Let $X_n$ a random variable of same law

    :confused: Hello I have a simple question : Let $X_n$ a random variable of same law If $V(X_n)\longrightarrow 0$ when $n\longrightarrow +\infty$ How schow that : $E(X_n)\longrightarrow C$ and $E(X_{n}^{2}\longrightarrow C^2$ and C is a constant? Thanks
  32. J

    Function of gaussian random variable

    I'm having trouble showing the following relation: E(exp(z)) = exp(E(z^2)/2) where z is a zero-mean gaussian variable and E() is the avg anyone can help?
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