(1) Expected Value(2) Moments(3) Variance, Standard Deviation and Coefficient of Variation(4) Moment Generating Function(5) Percentiles(6) Mode(7) Skewness and Kurtosis(8) Other useful resultsMath 370X - Lecture 4Expectation and Other Distribution ParametersSaumil PadhyaOctober 3, 2016(1) Expected ValueSymbol: E[X] or µ(i) Discrete Random VariablesE[X] =nXi=1xi· p(xi)E[g(X)] =nXi=1g(xi) · p(xi)(ii) Continuous Random VariablesE[X] =bZax · f(x)dxE[g(X)] =bZag(x) · f(x)dx(iii) PropertiesE[a] = a where a is a constantE[bX] = b · E[x] where b is a constantE[a + bX] = a + b · E[X] → expected value is a linear f unction(2) MomentsSymbol: E[Xn] → Raw MomentE[(X-µ)n] → Central Moment(i) Discrete Random Variables1E[Xn] =nXi=1xni· p(xi) → raw momentE[(X − µ)n] =nXi=1(xi− µ)n· p(xi) → central moment(ii) Continuous Random VariablesE[Xn] =bZaxn· f(x)dx → raw momentE[(X − µ)n] =bZa(x − µ)n· f(x)dx → central momentE[X] is the first raw moment i.e. E[Xn] when n=1(3) Variance, Standard Deviation and Coefficient of VariationSymbol: Var[X] or σ2→ VarianceSd[X] or σ → Standard DeviationCV[X] → Coefficient of VariationV ar[X] = σ2= E[(X − µ)2] = E[X2] − E[X]2Sd[X] = σ =pV ar[X]CV [X] =σµPropertiesV ar[X] ≥ 0V ar[a] = 0 where a is a constantV ar[bX] = b2· V ar[X] where a is a constantV ar[a + bX] = b2· V ar[X] → variance is not a linear f unction2(4) Moment Generating FunctionSymbol: MX(t) or M(t)(i) Discrete Random VariablesMX(t) = E[etx] =nXi=1etxi· p(xi)(ii) Continuous Random VariablesMX(t) =bZaetx· f(x)dx(iii) PropertiesMX(0) = 1M(n)X(0) = E[Xn]→ M(1)X(0) = E[X]→ M(2)X(0) = E[X2]d2dt2ln[MX(t)]t=0= V ar[X]Let Y = X1+ X2+ X3where X1, X2, X3are independent, thenMY(t) = MX1(t) · MX2(t) · MX3(t)(5) PercentilesSymbol: cpFor 0 < p < 1, 100pthpercentile of distribution of X is cpwhich satistfiesP [ X ≤ cp] ≥ pP [ X ≥ cp] ≥ 1 − pTo make it simpler we will always compute percentile cpby settingP [ X ≤ cp] = pMedian (midpoint) M is defined to be the 50th percentileP [ X ≤ c0.5] = P [X ≤ M ] = 0.53(6) ModeSymbol: Mode(X)Mode for distribution of X is any point(s) m for which f(x) is maximized.If X has a discrete distribution, mode m can be chosen as the x for which P[X=x] is maximized.If X has a continuous distribution, mode m is the point x for whichf(1)(x) = 0 andf(2)(x) < 0.(7) Skewness and KurtosisSymbol: Skew(X); Kurt(X)Skew(X) =E[(X − µ)3]σ3Kurt(X) =E[(X − µ)4]σ4(8) Other useful resultsE[X] =bZax · f(x)dx = a +bZa(1 − F (x))dx∞Z0tk· e−atdt =k!ak+1MX(t) =∞Xk=0tkk!· E[Xk] = 1 + t · E[X] +t22· E[X2] +t36· E[X3] +
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