Functions of Random Variables

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Functions of Random Variables
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Method of Distribution Functions

• X1,,Xn f(x1,,xn)
• Ug(X1,,Xn) Want to obtain fU(u)
• Find values in (x1,,xn) space where Uu
• Find region where Uu
• Obtain FU(u)P(Uu) by integrating f(x1,,xn)
  over the region where Uu
• fU(u) dFU(u)/du

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Example Uniform X

• Stores located on a linear city with density
  f(x)0.05 -10 x 10, 0 otherwise
• Courier incurs a cost of U16X2 when she delivers
  to a store located at X (her office is located at
  0)

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Example Sum of Exponentials

• X1, X2 independent Exponential(q)
• f(xi)q-1e-xi/q xigt0, qgt0, i1,2
• f(x1,x2) q-2e-(x1x2)/q x1,x2gt0
• UX1X2

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Method of Transformations

• XfX(x)
• Uh(X) is either increasing or decreasing in X
• fU(u) fX(x)dx/du where xh-1(u)
• Can be extended to functions of more than one
  random variable
• U1h1(X1,X2), U2h2(X1,X2), X1h1-1(U1,U2),
  X2h2-1(U1,U2)

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Example

• fX(x) 2x 0 x 1, 0 otherwise
• U10500X (increasing in x)
• x(u-10)/500
• fX(x) 2x 2(u-10)/500 (u-10)/250
• dx/du d((u-10)/500)/du 1/500
• fU(u) (u-10)/2501/500 (u-10)/125000 10
  u 510, 0 otherwise

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Method of Conditioning

• Uh(X1,X2)
• Find f(ux2) by transformations (Fixing X2x2)
• Obtain the joint density of U, X2
• f(u,x2) f(ux2)f(x2)
• Obtain the marginal distribution of U by
  integrating joint density over X2

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Example (Problem 6.11)

• X1Beta(a2,b2) X2Beta(a3,b1) Independent
• UX1X2
• Fix X2x2 and get f(ux2)

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Method of Moment-Generating Functions

• X,Y are two random variables
• CDFs FX(x) and FY(y)
• MGFs MX(t) and MY(t) exist and equal for
  tlth,hgt0
• Then the CDFs FX(x) and FY(y) are equal
• Three Properties
• YaXb ? MY(t)E(etY)E(et(aXb))ebtE(e(at)X)ebt
  MX(at)
• X,Y independent ? MXY(t)MX(t)MY(t)
• MX1,X2(t1,t2) Eet1X1t2X2 MX1(t1)MX2(t2) if
  X1,X2 are indep.

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Sum of Independent Gammas
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Linear Function of Independent Normals
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Distribution of Z2 (ZN(0,1))
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Distributions of and S2 (Normal data)
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Independence of and S2 (Normal Data)
Independence of TX1X2 and DX2-X1 for Case
of n2
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Independence of and S2 (Normal Data) P2
Independence of TX1X2 and DX2-X1 for Case
of n2
Thus TX1X2 and DX2-X1 are independent Normals
and S2 are independent
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Distribution of S2 (P.1)
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Distribution of S2 P.2
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Summary of Results

• X1,Xn random sample from N(m, s2) population
• In practice, we observe the sample mean and
  sample variance (not the population values m,
  s2)
• We use the sample values (and their
  distributions) to make inferences about the
  population values

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Order Statistics

• X1,X2,...,Xn ? Independent Continuous RVs
• F(x)P(Xx) ? Cumulative Distribution Function
• f(x)dF(x)/dx ? Probability Density Function
• Order Statistics X(1) X(2) ... X(n)
  (Continuous ? can ignore equalities)
• X(1) min(X1,...,Xn)
• X(n) max(X1,...,Xn)

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Order Statistics
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Example

• X1,...,X5 iid U(0,1)
• (iidindependent and identically distributed)

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Distributions of Order Statistics

• Consider case with n4
• X(1) x can be one of the following cases
• Exactly one less than x
• Exactly two are less than x
• Exactly three are less than x
• All four are less than x
• X(3) x can be one of the following cases
• Exactly three are less than x
• All four are less than x
• Modeled as Binomial, n trials, pF(x)

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Case with n4
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General Case (Sample of size n)
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Example n5 Uniform(0,1)
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