16sample - 2 Let x n ∼ N(0 1 0004 ⇔ E x n =...

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EECS 501 EXAMPLES OF SAMPLE FUNCTIONS Fall 2001 EX #1: To generate sample functions x ( n ) of a Bernoulli process with p=0.6: 1. Spin a wheel of fortune once , resulting in a number ω o [0 , 1). 2. Let ω o have decimal expansion 0 .w 1 w 2 w 3 w 4 w 5 ... where w i = 0 , 1 ... 9. 3. Set x ( n )=1 if w n =0,1,2,3,4 or 5 and x ( n )=0 if w n =6,7,8 or 9. ω o = 0.141592653589793 ... → { x ( n ) } = { 1,1,1,1,0,1,0,1,1,1,0,0,0,0,1 ... } . ω o = 0.718281828459045 ... → { x ( n ) } = { 0,1,0,1,0,1,0,1,0,1,1,0,1,1,1 ... } . 1. Different sequence of 0’s and 1’s associated with each ω o Ω=[0,1). Each sequence is a sample function of the Bernoulli random process. 2. Digits independent of each other (if ω o irrational); Pr[x(n)=1]=0.6. 3. Pr[ ω o irrational]=1 so this almost surely works (need precision). EX #2: Monte Carlo simulation of compounded rate of return on investments: 1. Let x ( n ) be the annual rate of return on investments in year 2000+n.
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Unformatted text preview: 2. Let x ( n ) ∼ N (0 . 1 , . 0004) ⇔ E [ x ( n )] = 10% (historical average). 3. Con²dence limits or intervals (see pp. 275-280 and recitation) are: Pr [8% < x ( n ) < 12%] = 67%; Pr [6% < x ( n ) < 14%] = 95%. 4. Let y ( n ) = p n i =1 (1+ x ( i )). Assume x ( n ) iidrvs. Then E [ y ( n )] = (1 . 1) n . 5. Generate 10 sample functions of x ( n ) (Matlab’s randn ) and y ( n ). Results plotted below. Curves are di±erent sample functions of y ( n ). • Note that the actual return can vary greatly from the expected return! So don’t plan on retiring too quickly! “Your performance may vary. ..” 10 20 30 40 20 40 60 #years invested since 2000 value of $1 invested...
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