14ma`ruza. Variance and Standard Deviation
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14ma`ruza.Variance and Standard Deviation
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 Properties of the Variance
 Quantiles
 Some Special Distributions
 The Polynomial Distribution
14ma`ruza.Variance and Standard DeviationThe expected value alone does not sufficiently characterize a random variable. We must consider also what deviation from the expected value can occur on average (see Sect. A.2.3.2). This dispersion is described by variance and standard deviation. Definition A.24 If μ is the expected value of a discrete realvalued random vari able X, then the value (provided that it exists) ∞ σ ^{2} = D^{2}(X) = E^{ }[X − μ]^{2}^{ }^{ }= ^{ }(x_{i} − μ)^{2}P (X = x_{i}) i=1 _{√}_{ }_{ } = =+ is called the variance, and the positive square root σ D(X) standard deviation of Xσ ^{2}^{ }is called the Let us again consider roulette as an example. If we bet m chips on a column, the variance is D (X) = 2m + _{37}_{ }m ^{·}^{ }37 ^{+} −m + _{37}_{ }m ^{·}^{ }37 ^{=}^{ }50653 ^{m} ≈ 1.97m . _{2} 1 ^{ }^{2 }^{ }12 1 ^{ }^{2 }^{ }25 99900 _{2 2} 35m + _{37}_{ }m · _{37}_{ }+ −m + _{37}_{ }m · _{37}_{ }= 50653 ^{m} ≈ 34.1m , Hence the standard deviation D(X) is about 1.40m. In comparison, the variance of a bet on a plain chance, that is, on a single number, has the same expected value, but the variance D (X) = _{2 }1 ^{ }^{2 } 1 1 ^{ }^{2 }^{ }36 1726272 _{2 2} 342 A Statistics and thus a standard deviation D(X) of about 5.84m. Despite the same expected value, the average deviation from the expected value is about 4 times as large for a bet on a plain chance than for a bet on a column. In order to define the variance of a continuous random variable, we only have to replace the sum by an integral—just as we did for the expected value. Definition A.25 If μ is the expected value of a continuous random variable X, then the value (provided that it exists) σ ^{2} = D^{2}(X) = ∫ ∞ (x − μ)^{2}f (x) dx −∞ _{√}_{ }_{ } = =+ is called the variance of X, and σ D(X) tion of X.
σ ^{2}^{ }is called the standard devia In this section we collect a few useful properties of the variance. Theorem A.13 Let X be a discrete random variable which takes no other values than a constant c. Then its variance is 0: σ ^{2} = D^{2}(X) = 0. Theorem A.14 Let X be a (discrete or continuous) realvalued random variable with variance D^{2}(X). Then the variance of the random variable Y = aX + b, a, b ∈ R, is D^{2}(Y ) = D^{2}(aX + b) = a^{2}D^{2}(X), and therefore, for the standard deviation, we have D(Y) = D(aX + b) = aD(X). The validity of this theorem (like the validity of the next theorem) can easily be checked by inserting the given expressions into the definition of the variance, once for discrete and once for continuous random variables. Theorem A.15 The variance σ ^{2} of a (discrete or continuous) realvalued random variable satisfies σ ^{2} = E^{ }X^{2}^{ }− μ^{2}. Theorem A.16 (variance of a sum of random variables, covariance) If X and Y are two (discrete or continuous) realvalued random variables, whose variances D^{2}(X) and D^{2}(Y ) exist, then D^{2}(X + Y) = D^{2}(X) + D^{2}(Y ) + 2[E(X · Y) − E(X) · E(Y)]. A.3 Probability Theory 343 · − · = [ − − ] The expression E(X Y) E(X) E(Y) E (X E(X))(Y E(Y)) is called the covariance of X and Y . From the (stochastic) independence of X and Y it follows that D^{2}(Z) = D^{2}(X + Y) = D^{2}(X) + D^{2}(Y ), that is, the covariance of independent random variables vanishes. Again the validity of this theorem can easily be checked by inserting the sum into the definition of the variance. By simple induction it can easily be generalized to finitely many random variables.
Quantiles are defined in direct analogy to the quantiles of a data set, with the frac tion of the data set replaced by the fraction of the probability mass. For continuous random variables, quantiles are often also called percentage points. Definition A.26 Let X be a realvalued random variable. Then any value x_{α}, 0 <α < 1, with P (X ≤ x_{α}) ≥ α and P (X ≥ x_{α}) ≥ 1 − α is called an αquantile of X (or of its distribution). Note that for discrete random variables, several values may satisfy both inequali ties, because their distribution function is piecewise constant. It should also be noted that the pair of inequalities is equivalent to the double inequality α − P (X = x) ≤ F_{X}(x) ≤ α, = where F_{X}(x) is the distribution function of a random variable X. For a continuous random variable X, it is usually more convenient to define that the αquantile is the value x that satisfies F_{X}(x) α. In this case a quantile can be computed from the inverse of the distribution function F_{X} (provided that it exists and can be specified in closed form).
In this section we study some special distributions, which are often needed in appli cations (see Sect. A.4 about inferential statistics).
Let X be a random variable that describes the number of trials of a Bernoulli exper iment of size n in which an event A occurs with probability p = P (A) in each trial. 344 A Statistics Then X has the distribution ∀x ∈ N : (x; P (X = x)) with P (X = x) = b_{X}(x; p, n) = p^{x}(1 − p)^{n}^{−}^{x} x also known as Bernoulli’s formula. The expression ^{n} =^{ }^{n}^{! }(pronounced “n n and is said to be binomially distributed with parameters p and n. This formula is choose x”) is called a binomial coefficient. The distribution satisfies the recursive relation x x!(n−x)! ∀x ∈ N_{0} : b_{X} (k 1 p, n) ^{(n}^{ }^{−}^{ }^{x)p }b + ; = X (x + 1)(1 − p) (x; p, n) with b_{X}(0; p, n) = (1 − p)^{n}. For the expected value and variance, we have μ = E(X) = np; σ ^{2} = D^{2}(X) = np(1 − p). n trials the event A_{i} , i = 1,...,k, occurs x_{i} times, k i=1 x_{i} = n, is equal to Bernoulli experiments can easily be generalized to more than two mutually exclu sive events. In this way one obtains the polynomial distribution, which is a multi dimensional distribution: a random experiment is executed independently n times. Let A_{1},..., A_{k} be mutually exclusive events, of which in each trial exactly one must occur, that is, let A_{1},..., A_{k} be an event partition. In every trial each event A_{i} oc curs with constant probability p_{i} = P (A_{i}), 1 ≤ i ≤_{.}k. Then the probability that in P (X_{1} = x_{1},...,X = x ) = ^{n }p^{x}^{1} ··· p^{xk} =^{ }^{n}^{! }p^{x}^{1} ··· p^{xk} . ^{k}^{ }^{k}^{ }x_{1} ... x_{k} ^{1} ^{k} x_{1}!··· x_{k}! ^{1 }^{k} = i 1 with parameters p_{1},..., p_{k} The bi_{ }nomial distribution is obviously a special case of for k = 2. The expression The total of all probabilities of all vectors (x_{1},..., x_{k}) with ^{.}^{k}^{ }x_{i} = n is called _{the} (kdimensional) polynomial distribution _{and} n. _{ } = n x_{1}...x_{k} n! x_{1}!···x_{k}_{ }! is called a polynomial coefficient, in analogy to the binomial x x!(n−x)! coefficient ^{ }^{n} =^{ }^{ }^{n}^{! }^{ }. Download 342.94 Kb. Do'stlaringiz bilan baham: 
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