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Text of the page (random words):
ega in b 12 this definition enables us to measure any subset b e displaystyle b in mathcal e in the target space by looking at its preimage which by assumption is measurable in more intuitive terms a member of ω displaystyle omega is a possible outcome a member of f displaystyle mathcal f is a measurable subset of possible outcomes the function p displaystyle p gives the probability of each such measurable subset e displaystyle e represents the set of values that the random variable can take such as the set of real numbers and a member of e displaystyle mathcal e is a well behaved measurable subset of e displaystyle e those for which the probability may be determined the random variable is then a function from any outcome to a quantity such that the outcomes leading to any useful subset of quantities for the random variable have a well defined probability when e displaystyle e is a topological space then the most common choice for the σ algebra e displaystyle mathcal e is the borel σ algebra b e displaystyle mathcal b e which is the σ algebra generated by the collection of all open sets in e displaystyle e in such case the e e displaystyle e mathcal e valued random variable is called an e displaystyle e valued random variable moreover when the space e displaystyle e is the real line r displaystyle mathbb r then such a real valued random variable is called simply a random variable note that we are not giving r displaystyle mathbb r the usual lebesgue σ displaystyle sigma algebra which is the completion of the borel σ displaystyle sigma algebra this choice allows for more measurable functions f ω r displaystyle f omega to mathbb r and makes it easier to check that a function f ω r displaystyle f omega to mathbb r is measurable as we only need to check that preimages of open sets are measurable real valued random variables edit in this case the observation space is the set of real numbers recall ω f p displaystyle omega mathcal f p is the probability space for a real observation space the function x ω r displaystyle x colon omega rightarrow mathbb r is a real valued random variable if ω x ω r f r r displaystyle omega x omega leq r in mathcal f qquad forall r in mathbb r this definition is a special case of the above because the set r r r displaystyle infty r r in mathbb r generates the borel σ algebra on the set of real numbers and it suffices to check measurability on any generating set here we can prove measurability on this generating set by using the fact that ω x ω r x 1 r displaystyle omega x omega leq r x 1 infty r moments edit the probability distribution of a random variable is often characterised by a small number of parameters which also have a practical interpretation for example it is often enough to know what its average value is this is captured by the mathematical concept of expected value of a random variable denoted e x displaystyle operatorname e x and also called the first moment in general e f x displaystyle operatorname e f x is not equal to f e x displaystyle f operatorname e x once the average value is known one could then ask how far from this average value the values of x displaystyle x typically are a question that is answered by the variance and standard deviation of a random variable e x displaystyle operatorname e x can be viewed intuitively as an average obtained from an infinite population the members of which are particular evaluations of x displaystyle x mathematically this is known as the generalised problem of moments for a given class of random variables x displaystyle x find a collection f i displaystyle f_ i of functions such that the expectation values e f i x displaystyle operatorname e f_ i x fully characterise the distribution of the random variable x displaystyle x moments can only be defined for real valued functions of random variables or complex valued etc if the random variable is itself real valued then moments of the variable itself can be taken which are equivalent to moments of the identity function f x x displaystyle f x x of the random variable however even for non real valued random variables moments can be taken of real valued functions of those variables for example for a categorical random variable x that can take on the nominal values red blue or green the real valued function x green displaystyle x text green can be constructed this uses the iverson bracket and has the value 1 if x displaystyle x has the value green 0 otherwise then the expected value and other moments of this function can be determined functions of random variables edit a new random variable y displaystyle y can be defined by applying a real borel measurable function g r r displaystyle g colon mathbb r rightarrow mathbb r to the outcomes of a real valued random variable x displaystyle x that is y g x displaystyle y g x the cumulative distribution function of y displaystyle y is then f y y p g x y displaystyle f_ y y operatorname p g x leq y if function g displaystyle g is invertible i e h g 1 displaystyle h g 1 exists where h displaystyle h is g displaystyle g s inverse function and is either increasing or decreasing then the previous relation can be extended to obtain f y y p g x y p x h y f x h y if h g 1 increasing p x h y 1 f x h y if h g 1 decreasing displaystyle f_ y y operatorname p g x leq y begin cases operatorname p x leq h y f_ x h y text if h g 1 text increasing operatorname p x geq h y 1 f_ x h y text if h g 1 text decreasing end cases with the same hypotheses of invertibility of g displaystyle g assuming also differentiability the relation between the probability density functions can be found by differentiating both sides of the above expression with respect to y displaystyle y in order to obtain 10 f y y f x h y d h y d y displaystyle f_ y y f_ x bigl h y bigr left frac dh y dy right if there is no invertibility of g displaystyle g but each y displaystyle y admits at most a countable number of roots i e a finite or countably infinite number of x i displaystyle x_ i such that y g x i displaystyle y g x_ i then the previous relation between the probability density functions can be generalized with f y y i f x g i 1 y d g i 1 y d y displaystyle f_ y y sum _ i f_ x g_ i 1 y left frac dg_ i 1 y dy right where x i g i 1 y displaystyle x_ i g_ i 1 y according to the inverse function theorem the formulas for densities do not demand g displaystyle g to be increasing in the measure theoretic axiomatic approach to probability if a random variable x displaystyle x on ω displaystyle omega and a borel measurable function g r r displaystyle g colon mathbb r rightarrow mathbb r then y g x displaystyle y g x is also a random variable on ω displaystyle omega since the composition of measurable functions is also measurable however this is not necessarily true if g displaystyle g is lebesgue measurable citation needed the same procedure that allowed one to go from a probability space ω p displaystyle omega p to r d f x displaystyle mathbb r df_ x can be used to obtain the distribution of y displaystyle y example 1 edit let x displaystyle x be a real valued continuous random variable and let y x 2 displaystyle y x 2 f y y p x 2 y displaystyle f_ y y operatorname p x 2 leq y if y 0 displaystyle y 0 then p x 2 y 0 displaystyle p x 2 leq y 0 so f y y 0 if y 0 displaystyle f_ y y 0 qquad hbox if quad y 0 if y 0 displaystyle y geq 0 then p x 2 y p x y p y x y displaystyle operatorname p x 2 leq y operatorname p x leq sqrt y operatorname p sqrt y leq x leq sqrt y so f y y f x y f x y if y 0 displaystyle f_ y y f_ x sqrt y f_ x sqrt y qquad hbox if quad y geq 0 example 2 edit suppose x displaystyle x is a random variable with a cumulative distribution f x x p x x 1 1 e x θ displaystyle f_ x x p x leq x frac 1 1 e x theta where θ 0 displaystyle theta 0 is a fixed parameter consider the random variable y l o g 1 e x displaystyle y mathrm log 1 e x then f y y p y y p l o g 1 e x y p x l o g e y 1 displaystyle f_ y y p y leq y p mathrm log 1 e x leq y p x geq mathrm log e y 1 the last expression can be calculated in terms of the cumulative distribution of x displaystyle x so f y y 1 f x log e y 1 1 1 1 e log e y 1 θ 1 1 1 e y 1 θ 1 e y θ displaystyle begin aligned f_ y y 1 f_ x log e y 1 5pt 1 frac 1 1 e log e y 1 theta 5pt 1 frac 1 1 e y 1 theta 5pt 1 e y theta end aligned which is the cumulative distribution function cdf of an exponential distribution example 3 edit suppose x displaystyle x is a random variable with a standard normal distribution whose density is f x x 1 2 π e x 2 2 displaystyle f_ x x frac 1 sqrt 2 pi e x 2 2 consider the random variable y x 2 displaystyle y x 2 we can find the density using the above formula for a change of variables f y y i f x g i 1 y d g i 1 y d y displaystyle f_ y y sum _ i f_ x g_ i 1 y left frac dg_ i 1 y dy right in this case the change is not monotonic because every value of y displaystyle y has two corresponding values of x displaystyle x one positive and negative however because of symmetry both halves will transform identically i e f y y 2 f x g 1 y d g 1 y d y displaystyle f_ y y 2f_ x g 1 y left frac dg 1 y dy right the inverse transformation is x g 1 y y displaystyle x g 1 y sqrt y and its derivative is d g 1 y d y 1 2 y displaystyle frac dg 1 y dy frac 1 2 sqrt y then f y y 2 1 2 π e y 2 1 2 y 1 2 π y e y 2 displaystyle f_ y y 2 frac 1 sqrt 2 pi e y 2 frac 1 2 sqrt y frac 1 sqrt 2 pi y e y 2 this is a chi squared distribution with one degree of freedom example 4 edit suppose x displaystyle x is a random variable with a normal distribution whose density is f x x 1 2 π σ 2 e x μ 2 2 σ 2 displaystyle f_ x x frac 1 sqrt 2 pi sigma 2 e x mu 2 2 sigma 2 consider the random variable y x 2 displaystyle y x 2 we can find the density using the above formula for a change of variables f y y i f x g i 1 y d g i 1 y d y displaystyle f_ y y sum _ i f_ x g_ i 1 y left frac dg_ i 1 y dy right in this case the change is not monotonic because every value of y displaystyle y has two corresponding values of x displaystyle x one positive and negative differently from the previous example in this case however there is no symmetry and we have to compute the two distinct terms f y y f x g 1 1 y d g 1 1 y d y f x g 2 1 y d g 2 1 y d y displaystyle f_ y y f_ x g_ 1 1 y left frac dg_ 1 1 y dy right f_ x g_ 2 1 y left frac dg_ 2 1 y dy right the inverse transformation is x g 1 2 1 y y displaystyle x g_ 1 2 1 y pm sqrt y and its derivative is d g 1 2 1 y d y 1 2 y displaystyle frac dg_ 1 2 1 y dy pm frac 1 2 sqrt y then f y y 1 2 π σ 2 1 2 y e y μ 2 2 σ 2 e y μ 2 2 σ 2 displaystyle f_ y y frac 1 sqrt 2 pi sigma 2 frac 1 2 sqrt y e sqrt y mu 2 2 sigma 2 e sqrt y mu 2 2 sigma 2 this is a noncentral chi squared distribution with one degree of freedom some properties edit the probability distribution of the sum of two independent random variables is the convolution of each of their distributions probability distributions are not a vector space they are not closed under linear combinations as these do not preserve non negativity or total integral 1 but they are closed under convex combination thus forming a convex subset of the space of functions or measures equivalence of random variables edit there are several different senses in which random variables can be considered to be equivalent two random variables can be equal equal almost surely or equal in distribution in increasing order of strength the precise definition of these notions of equivalence is given below equality in distribution edit if the sample space is a subset of the real line random variables x and y are equal in distribution denoted x d y displaystyle x stackrel d y if they have the same distribution functions p x x p y x for all x displaystyle operatorname p x leq x operatorname p y leq x quad text for all x to be equal in distribution random variables need not be defined on the same probability space two random variables having equal moment generating functions have the same distribution this provides for example a useful method of checking equality of certain functions of independent identically distributed iid random variables however the moment generating function exists only for distributions that have a defined laplace transform almost sure equality edit two random variables x and y are equal almost surely denoted x a s y displaystyle x stackrel text a s y if and only if the probability that they are different is zero p x y 0 displaystyle operatorname p x neq y 0 for all practical purposes in probability theory this notion of equivalence is as strong as actual equality it is associated to the following distance d x y ess sup ω x ω y ω displaystyle d_ infty x y operatorname ess sup _ omega x omega y omega where ess sup represents the essential supremum in the sense of measure theory equality edit finally the two random variables x and y are equal if they are equal as functions on their measurable space x ω y ω for all ω displaystyle x omega y omega qquad hbox for all omega this notion is typically the least useful in probability theory because in practice and in theory the underlying measure space of the experiment is rarely explicitly characterized or even characterizable practical difference between notions of equivalence edit since we rarely explicitly construct the probability space underlying a random variable the difference between these notions of equivalence is somewhat subtle essentially two random variables considered in isolation are practically equivalent if they are equal in distribution but once we relate them to other random variables defined on the same probability space then they only remain practically equivalent if they are equal almost surely for example consider the real random variables a b c and d all defined on the same probability space suppose that a and b are equal almost surely a a s b displaystyle a stackrel text a s b but a and c are only equal in distribution a d c displaystyle a stackrel d c then a d a s b d displaystyle a d stackrel text a s b d but in general a d c d displaystyle a d neq c d not even in distribution similarly we have that the expectation values e a d e b d displaystyle mathbb e ad mathbb e bd but in general e a d e c d displaystyle mathbb e ad neq mathbb e cd therefore two random variables that are equal in distribution but not equal almost surely can have different covariances with a third random variable convergence edit main article convergence of random variables a significant theme in mathematical statistics consists of obtaining convergence results for certain sequences of random variables for instance the law of large numbers and the central limit theorem there are various senses in which a sequence x n displaystyle x_ n of random variables can converge to a random variable x displaystyle x these are explained in the article on convergence of random variables see also edit mathematics portal aleatoricism algebra of random variables event probability theo...
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