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. 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?

    Professor Roberto has to take an oral examination. The grading scale is as follows: 5: = best and 1: = worst. At most he only gives the note 4. Each student under review is questioned if he is a Lakers fan. The student's grade is based on his answer (is a fan / not a fan) and on the language in...
  4. G

    Function of random variable, limits of integration

    Homework Statement X is uniformly distributed over [-1,1]. Compute the density function f(y) of Y = 2X2 + 1. Homework Equations The Attempt at a Solution FY(Y) = P(Y < y) = P(2X2 + 1 < y) = P(X < +\sqrt{1/2(y-1)} = FX(+\sqrt{1/2(y-1)}) We have that f(x) = 0.5 for -1 < x <...
  5. twoski

    Generating a Random Variable with a Specific Distribution Function

    Homework Statement Give a method for generating a random variable with distribution function F(x) = 1/2(x+x^{2}) 0<x<1 The Attempt at a Solution From what i can tell i am supposed to do something like: Let U be a uniformly distributed random variable over (0,1). U =...
  6. D

    What is the Cumulative Distribution Function for a Continuous Random Variable?

    The cumulative distribution function of a continuous random variable is given as follows: 0 0 ( ) 0 5 5 1 5 X if x x F x if x x           a. Determine and name the density function of . [02] b. Use both and ( ) X F x to find P(X  3) . [05] c. Find the variance of ...
  7. L

    Showing a random Variable has a continuous uniform distribution

    f(x)=1, θ-1/2 ≤ x ≤ θ+1/2 Given that Z=(b-a)(x-θ)+(1/2)(a+b) how would you show that Z has a continuous uniform distribution over the interval (a,b)? Any help would be much appreciated.
  8. twoski

    Solving Normal Random Variable Equations for P(X(X-1) > 2) and P(|X| > a)

    Homework Statement X is a normal random variable with mean 1, variance 4. 1. Find P( X(X-1) > 2 ) 2. Find a value 'a' for which P(|X| > a ) = .25 The Attempt at a Solution I had no idea how to start 1. For 2, i got this far then got stuck: P(|X| > a) = 1 - P((X-1)/2 <=...
  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

    Homework Statement A trial consists of throwing two dice. The result is counted as successful if the sum of the outcomes is 12. What is the probability that the number of successes in 36 such trials is greater than one? What is this probability if we approximate its value using the Poisson...
  11. A

    Cdf of a discrete random variable and convergence of distributions

    In the page that I attached, it says "...while at the continuity points x of F_x (i.e., x \not= 0), lim F_{X_n}(x) = F_X(x)." But we know that the graph of F_X(x) is a straight line y=0, with only x=0 at y=1, right? But then all the points to the right of zero should not be equal to the limit of...
  12. W

    Expected Value and Binomial Random Variable

    1. In scanning electron microscopy photography, a specimen is placed in a vacuum chamber and scanned by an electron beam. Secondary electrons emitted from the specimen are collected by a detector and an image is displayed on a cathode-ray tube. This image is photographed. In the past a 4- ...
  13. O

    MHB Discrete-continuous random variable

    Hello everyone! I'm looking at the following random variables: $Z_1$ is normally distributed with zero mean and variance $\sigma _1 ^2$ $Z_2$ is normally distributed with zero mean and variance $\sigma _2 ^2$ $B$ is Bernoulli with probability of success $p$. $X$ is a random variable that...
  14. K

    Conditional Expectation of a random variable

    My professor made a rather concise statement in class, which sums to this: E(Y|X=xi) = constant. E(Y|X )= variable. Could anyone help me understand how the expectation is calculated for the second case? I understand that for different values of xi, we'll have different values for the...
  15. C

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

    Homework Statement 1000 independent rolls of a fair die will be made. Compute an approximation to the probability that the number 6 will appear between 150 and 200 times inclusively. If the number 6 appears exactly 200 times, find the probability that the number 5 will appear less than 150...
  16. S

    Continuous Functions of One Random Variable

    My problem is as follows (sorry, but the tags were giving me issues. I tried to make it as readable as possible): Let X have the pdf f(x)= θ * e-θx, 0 < x < ∞ Find pdf of Y = ex I've gone about this the way I normally do for these problems. I have G(y) = P(X < ln y) = ∫ θ * e-θx...
  17. U

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

    Work done so far... Integrating from 0 to infinity and equating it to 1, we get (c/2*10^-3) = 1 c= 2/1000 =0.002 Is it correct? http://www.chegg.com/homework-help/questions-and-answers/-q3136942#
  18. W

    Is a random variable really random or just a function?

    lets say that X is some random variable that takes +1 if rational otherwise -1. at http://tutorial.math.lamar.edu/Classes/CalcI/TheLimit.aspx in example 4, can we consider g(x) as a random variable because it's behaviour is same, right? is random variable really random or just function? I found...
  19. L

    Variance of Continous Random Variable

    I have the continuous random variable Y, defined such that: Y=3X+2 and the PDF of x is zero everywhere but: f(x)=\frac{1}{4}e^{\frac{-x}{4}}, x>0 I correctly got the mean like so: \mu=E(h(x))=\int^{\infty}_{0} h(x)f(x) and evaluated it to be 10. I am unsure as to how I go about...
  20. F

    Basic random variable question - measure theory approach

    I have always struggled in understanding probability theory, but since coming across the measure theoretic approach it seems so much simpler to grasp. I want to verify I have a couple basic things.So say we have a set χ. Together with a σ-algebra κ on χ, we can call (χ,κ) a measurable space...
  21. denjay

    Most probable value of a random variable?

    A problem in this book asks for the most probable value of a random variable x. As far as I know, if a random variable has "most probable value" then it isn't a random variable. The problem is attached. It is the second question in part b. Could the answer be that there is no most probable...
  22. E

    The distribution of function of random variable

    i have this question i do find the distribution like this figure : and i plot the y like this: now i want to find the distribution of y i tried to take the distribution for each interval in Fx(x) like this : but the solution in the book said : who is wrong me or the book...
  23. S

    Statistics-Probability Distribution of Discrete Random Variable

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  24. W

    What is a sequence of random variable?

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  25. P

    Transformation of random variable

    Hi there, I am currently reading Rohatgi's book "An introduction to probabilty and statistics" (http://books.google.de/books?id=IMbVyKoZRh8C&lpg=PP1&hl=de&pg=PA62#v=onepage&q&f=true). My questions concerns the "technique" of finding the PDF of a transformed random varibale Y by a function...
  26. J

    Division of Chi Squared Random Variables

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  27. fluidistic

    Characteristic function of a continuous random variable

    Homework Statement I must calculate the characteristic function as well as the first moments and cumulants of the continuous random variable f_X (x)=\frac{1}{\pi } \frac{c}{x^2+c^2} which is basically a kind of Lorentzian.Homework Equations The characteristic function is simply a Fourier...
  28. C

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

    Homework Statement Let f(x)=x/8 be the density of X on [0,4], zero elsewhere. a) Show that f(x) is a valid density and compute E(X) b) Define Y=1/X. Calculate E(Y) c) Determine the density function for Y The Attempt at a Solution a) is just really basic. I've solved that one. b)...
  29. lahanadar

    Random Process vs Random Variable vs Sample Space

    Hi everybody, I try to figure out connections and differences between random variables (RV), random processes (RP), and sample spaces and have confusions on some ideas you may want to help me. All sources I searched says that RP assigns each element of a sample space to a time function. I want...
  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

    EDIT: Oh and I forgot that $p_Y(y) = 0$ otherwise.
  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)

    Homework Statement In the winter, the monthly demand in tonnes, for solid fuel from a coal merchant may be modeled by the continuous random variable X with probability density function given by: f(x)=\frac{x}{30} 0≤x<6 f(x)=\frac{(12-x)^{2}}{180} 6≤x≤12 f(x)=0 otherwise (a)...
  36. K

    Limits for a truncated random variable

    Suppose that X is a random variable distributed in the interval [a;b] with pdf f(x) and cdf F(x). Clearly, F(b)=1. I only observe X for values that are bigger than y. I know that E(X|X>y)=\frac{\int_y^b xf(x)dx}{1-F(y)}. Moreover, \frac{∂E(X|X>y)}{∂y}=\frac{f(y)}{1-F(y)}[E(X|X>y)-y] I...
  37. T

    Distribution of sum of discrete random variable

    Edit: I have to think more about this, I'll post later.
  38. T

    Probability - Poisson Random Variable?

    1. Homework Statement During a typical Pennsylvania winter, I80 averages 1.6 potholes per 10 miles. A certain county is responsible for repairing potholes in a 30 mile stretch of the interstate. Let X denote the number of potholes the county will have to repair at the end of next winter. 1...
  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

    How to Derive Upper and Lower Bounds for a 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

    Any function is not a Random Variable

    There are plenty example of functions are random variables from my class note. I only interested of thinking up functions are not random variables. If you know functions are not random variables please please reply this post. This class is about set theory, probability measure, Borel...
  45. N

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

    Homework Statement Show that if X is a bounded random variable, then E(X) exists.Homework Equations The Attempt at a Solution I am having trouble of finding out where to begin this proof.This is what I got so far: Suppose X is bounded. Then there exists two numbers a and b such that P(X > b)...
  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

    Homework Statement A random variable X has probability generating function gX(s) = (5-4s2)-1 Calculate P(X=3) and P(X=4) Homework Equations The Attempt at a Solution Ehh don't really know where to go with one... I know: gX(s) = E(sx) = Ʃ p(X=k)(sk) Nit sure how to proceed.. Any help would...
  49. T

    Distribution of Bernoulli random variable

    Homework Statement a) Let X1, X2, ...XN be a collection of independent Bernoulli random variables. What is the distribution of Y = \sumNi = 1 Xi b) Show E(Y) = np Homework Equations Bernoulli equations f(x) = px(1-p)1-x The Attempt at a Solution a)X1 + X2 + ... + XN = p...
  50. M

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

    Determine constant c so that random variable will have a t distribution? Homework Statement Suppose that five random variables x1, x2, x3, x4, x5 are independent and have normal distribution N(0,1). Determine a constant c such that the random variable c*(x1+x2)/\sqrt{x_3^2 + x_4^2 +...
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