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Evaluating E(X) and Var X by moment generating function

Evaluating E(X) and Var X by moment generating function. Mysterious Mathematics Ahead! Student Discretion Advised. Xijin Ge SDSU Stat/Math. By Definition:. E(X) for a geometric r.v. Both sides multiply by q :. Subtract above equations:. Geometric Series!. By Definition:.

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Evaluating E(X) and Var X by moment generating function

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  1. Evaluating E(X) and Var X by moment generating function Mysterious Mathematics Ahead! Student Discretion Advised. Xijin Ge SDSU Stat/Math

  2. By Definition: E(X) for a geometric r.v. Both sides multiply by q: Subtract above equations: Geometric Series!

  3. By Definition: Var X for a geometric r.v. ? ? ? ?

  4. Moment Generating Function provides an “easier” method to calculate E(x), E(x^2) etc 1st ordinary moment 2nd ordinary moment If we can work out a mathematical expression for the m.g.f., then we could take derivatives to obtain the ordinary moments K-th ordinary moment

  5. Proof:

  6. M.G.F. for geometric distribution Geometric series

  7. Using M.G.F. to calculate E(X) for geometric distribution Its first derivative: Evaluating this derivative at t=0:

  8. Using M.G.F. to calculate Variance for geometric distribution Second order derivative: Evaluating it at t=0:

  9. Geometric Distribution Cumulative distribution function: Moment generating function:

  10. M.G.F.’s are wonderful: • Completely identifies a distribution. • If a distribution has a m.g.f., then it is unique. • If a r.v. has a m.g.f.: then it follows a geometric distribution with p=0.4

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