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.
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...
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}##...
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...
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...
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...
Hi all, I have a problem on linear estimation that I would like help on. This is related to Wiener filtering.
Problem:
I attempted part (a), but not too sure on the answer. As for unconstrained case in part (b), I don't know how to find the autocorrelation function, I applied the definition...
I've came across the two following theorems in my studies of Probability Generating Functions:
Theorem 1:
Suppose ##X_1, ... , X_n## are independent random variables, and let ##Y = X_1 + ... + X_n##. Then,
##G_Y(s) = \prod_{i=1}^n G_{X_i}(s)##
Theorem 2:
Let ##X_1, X_2, ...## be a sequence of...
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...
Hello all, sorry for the large wall of text but I'm really trying to understanding a problem from a study guide. I am quite unsure on how to approach the following multi-part problem. Any help would be appreciated.
Problem:
Useful references I'm using to attempt the problem
My attempt:
For...
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...
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:
$$\displaystyle \int_{-\infty }^{\infty }\!1/2\,{\frac {...
Hey! :giggle:
What does it mean to give the mapping for a random variable? Do we have to give the outcome space and the probability function? Does it hold that $X: ( \Omega, P)\mapsto \mathbb{R}$ ? :unsure:
(a)
$$\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...
Hey! 😊
A multiple choice test consists of 10 questions. For every question there are five possible answers, of which exactly one is correct. A test candidate answers all questions by chance.
(a) Give a suitable random variable with value range and probability distribution in order to work on...
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You participate in the following game :
You toss a fair coin until heads falls, but no more than three times. You have to pay $1$ euro for each throw. If your head falls, you win $3$ euros. The random variable $X$ describes your net profit (profit
minus stake). Give the values that $X$...
I want to develop a 2D random field and its change with time with constant velocity. My process:
1. Define a 2D grid [x, y] with n \times n points
2. Define 1D time axis [t] with n_t elements
3. Find the lagrangian distance between the points in space with the velocity in x and y ...
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...
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
I have a series of variables X i where ultimately the variables Xi each follow approximately an exponential distribution with a constant rate. In the beginning, there is a certain long-term trend. Is there a probability model in which Xi depends on the outcome of Xi-1 so that in the long run...
Given an Exponentially Distributed Random Variable $X\sim \exp(1)$, I need to find $\mathbb{E}[P_v]$, where $P_v$ is given as:$$ P_v=
\left\{
\begin{array}{ll}
a\left(\frac{b}{1+\exp\left(-\bar \mu\frac{P_s X}{r^\alpha}+\varphi\right)}-1\right), & \text{if}\ \frac{P_s X}{r^\alpha}\geq P_a,\\
0...
Let $a,b,c, \tau$ be positive constants and $x$ is an exponentially distributed variable with parameter $\lambda = 1$, i.e. $x\sim\exp(1)$.
\begin{equation}
E = \tau\Big[a\frac{1+a}{1+e^{-bx+c}} - 1 \Big]^+
\end{equation}
where $[z]^+ = \max(z,0)$
How can I find
The PDF for $E$
The...
I am not sure about how to approach this.
Since the volume is uniformly distributed, the mean volume is ##(5.7+5.1)/2=5.4##, which is less than ##5.5##. From this, I could say that, on average, the producer won't spend any extra dollars.
But then I thought that maybe I should interpret this as...
I've read that we can define a random variable on a probability space ##(\Omega, F, P)## such that it is a function that maps elements of the sample space to a measurable space - for instance, the reals - i.e. ##X: \Omega \rightarrow \mathbb{R}##.
That being said, it's often treated (at least...
My question is as follows. In the attached paper a formula is given on page 272 for the expectation of Tn (formula 23) and for the variance of Tn (formula 24). Now I would like to know what the formulas look like for Tn 's third and fourth central moment.
$X_1, X_2, ..., X_{15}$ are independently to each other and follow $N (7, 3^2)$ what distribution the following statistics follow$T = \frac{(\bar{X}− 7)}{\sqrt{s^2/15}}$i know this follow t distribution $t_(n-1) =t_{14}$but how do i find what distribution $T^2$ follows, can i just multiply it?$T...
Hello,
A discrete random variable X takes values $x_1,...,x_n$ each with probability $\frac1n$. Let Y=g(X) where g is an arbitrary real-valued function. I want to express the probability function of Y(pY(y)=P{Y=y}) in terms of g and the $x_i$
How can I answer this question?
If any member...
Quantum Electrodynamics (QED) has some observable effects such as the lamb shift, which is mainly caused by the vacuum polarization and the electron self-energy. These effects contribute to the "smearing" of the electron in an unpredictable manner, other than the uncertainty we already have...
The following is related to Poisson process:
$$P(N_1=2, N_4=6) = P(N_1=2, N_4-N_1=4) = P(N_1=2) \cdot P(N_3=4)$$
Why is $$(N_3=4)=(N_4-N_1=4)$$?
Can anyone explain?
Homework Statement
Let X be a random variable. It is not specified if it is continuous or discrete. Let g(x) alway positive and strictly increasing. Deduce this inequality:
$$P(X\geqslant x) \leqslant \frac{Eg(X)}{g(x)} \: $$
where x is real.
Homework Equations
I know that if X is discrete...
Problem:
We play roulette in a casino. We watch 100 rounds that result in a number between 1 and 36. and count the number of rounds for which the result is odd.
assuming that the roulette is fair, calculate the mean and deviation
Solution:
I understand that the probability - Pr = 0.5. and...
If we have a series of, say, twenty coin tosses, then each discernable specific series of outcomes has equal probability to occur. However, there is only one discernable specific series consisting of twenty 1's, while there are many more discernable series consisting of ten 1's and ten 0's.
So...
Hi,
Lets say I have N independent, not necessarily identical, random variable. I define a new random variable as
$$Y=Σ^{N}_{i=0} X_{i}$$
does Y follow a normalized probability distribution?
My textbook says that if ##X: \Omega \to \mathbb{R}## is discrete stochast (I.e., there are only countably many values that get reached), then it suffices to know the probability function ##p(x) = \mathbb{P}\{X =x\}## in order to know the distribution function ##\mathbb{P}_X: \mathcal{R} \to...
Hey! :o
For a random variable $X$ the skewness is defined by \begin{equation*}\eta (X):=E\left (\left (\frac{X-\mu }{\sigma}\right )^3\right )\end{equation*} where $E(X)=\mu$ and $\text{Var}(X)=\sigma^2$.
I want to show that \begin{equation*}\eta...
Homework Statement
X is a Poisson Random Variable with rate of 1 per hour, following the Poisson arrival process
a. Find the probability of no arrivals during a 10 hour interval
b. Find the probability of X > 10 arrivals in 2 hours
c. Find the average interarrival time.
d. For an interval of 2...
Homework Statement
X is a geometric random variable with p = 0.1. Find:
##a. F_X(5)##
##b. Pr(5 < X \leq 11)##
##c. Pr(X=7|5<X\leq11)##
##d. E(X|3<X\leq11)##
##e. E(X^2|3<X\leq11)##
##f. Var(X|3<X\leq11)##Homework EquationsThe Attempt at a Solution
Can someone check my work and help me?
a...
Homework Statement
See attached image (See below)
Homework Equations
Differential equations.
And a combination of discrete & continuous distributions
The Attempt at a Solution
The Continous Distribution Function (CDF) is given in the question. So I differentiated it with respect to x...
I just have a couple of questions about how it can be zero probability.
In case, you have a continuous cumulative probability distribution such that there is a derivative at each point not equal to zero. This means that every point as a different value than the other which means that every...
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?
Working through a paper about whose rigor I have my doubts, but I am always glad to be corrected. In the paper I find the following:
"We now investigate the random variable q. There are the following restrictions on q:
1) The variable q must characterize a stochastic process in the test...
On the attachment, I was told my joint pdf was right, but the support was NOT 0<y1y2<1 0<y2<1, so maybe it's right now?
Obviously B and C are incorrect, too, since they don't integrate to 1.
I'm probably making just a few simple mistakes. Thanks in advance!
Homework Statement
Suppose that the number of asbestos particles in a sam-
ple of 1 squared centimeter of dust is a Poisson random variable
with a mean of 1000. What is the probability that 10 squared cen-
timeters of dust contains more than 10,000 particles?
Homework Equations
E(aX+b) =...
Hello, could anyone please explain me some logic or derivation behind the approximation:
Found it in the Hull Derivatives book without further explanation. Thanks for help
Homework Statement
Consider the bivariate vector random variable ##(X,Y)^T## which has the probability density function $$f_{X,Y}(x,y) = \theta xe^{-x(y+\theta)}, \quad x\geq 0, y\geq 0 \; \; \text{and} \; \; \theta > 0.$$
I have shown that the marginal distribution of ##X## is ##f_X(x|\theta)...