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# stat400lec9 - Probability of success p Probability of...

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Statistics 400 Section 3.2 3.3 Expected value and Bernoulli Trials Review: Expected value of u(X) E[u(X)]= ) ( ) ( x f x u =u(x1)*f(x1)+u(x2)*f(x2)+ …+ u(xk)*f(xk) Example: Let X has p.m.f f(x)= 10 x , x=1,2,3,4 Calculate E[X], E[X 2 ] and E[X(5-X)] Ping Ma Lecture 9 Fall 2005 - 1 -

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Properties of Mathematical Expectation (a) E[c]=c if c is a constant. (b) E[cu(X)]=cE[u(X)] (c) E[c1*u1(X)+c2*u2(X)]=c1*E[u1(X)]+c2*E[u2(X)] Mean: ] [ X E = μ Variance: ] ) [( 2 2 σ - = X E = ] 2 [ 2 2 + - X X E Let X be the outcome when rolling a fair die. Calculate mean and variance of X. Ping Ma Lecture 9 Fall 2005 - 2 -
Let r be a positive integer, E[X r ]= ) ( x f x r is called rth moment of the distribution about the origin. E[(X-b) r ]= - ) ( ) ( x f b x r is called rth moment of the distribution about b. Bernoulli Experiment: A Bernoulli experiment is a random experiment, the outcome of which can be classified as success and failure. Example : tossing a coin

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Unformatted text preview: Probability of success p Probability of failure q=1-p. Let X be random variable of Bernoulli distribution. Outcome Success 1 Failure Probability p q=1-p X can take value of either 0 or 1 Notation: X ~ Ber (p) Setting: 1. Each observation falls into one of two categories: Success or Failure 2. P (Success) = p Probability mass function is f(x)= Ping Ma Lecture 9 Fall 2005 - 3 -Expected value Variance A sequence of Bernoulli Trials: A sequence of Bernoulli experiments are conducted independently and the probability of success (p) remains the same. Example: For a million of instant lottery tickets, suppose that 205 are winner, If 5 tickets are purchased, what is the probability we have (0,0,0,1,0) Ping Ma Lecture 9 Fall 2005 - 4 -Ping Ma Lecture 9 Fall 2005 - 5 -...
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stat400lec9 - Probability of success p Probability of...

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