E[kX] = kμ E[X+c] = μ+c E[kX+c] = kμ+c Var(kX)=k2σ2 Var(X+c)=σ2 Var(kX+c)=k2σ2 ??? SD(kX)=√k X no? CLT 4 coins, #H 0 1 2 3 4 .0625 .25 .375 .25 .0625 4 coins, #H-#T 0 2 -2 4 -4 3/8 1/4 1/4 1/16 1/16 5 coins, #H 0 1 2 3 4 5 .03125 .15625 .3125 .3125 .15625 .03125 scores 0 1 2 3 4 5 .1 .1 .2 .4 .1 .1 Lottery/raffle: Expected value of ticket= (payoff-price)*chance_of_winning + -price*(1-chance_of_winning) x P(x) ----- ----------- payoff-price prob. of winning -price prob. of losing = 1-prob.win Price: $10 Payoff: $15000 #tickets:2000 --> chance of winning: 1/2000 14990 -10 .0005 .9995 mu= -$2.50 Pick 3 $1 ticket, $500 payout, 1/1000 chance of winning 499 -1 .001 .999 mu= -$.50 n Multiple payouts: Each (payout-price)*chance_winning + ... + -price*(1-n*chance_of_winning) Price:$10. Payouts:$15000, $210, $110 Chance of winning any: 1/2000 14990 100 200 -10 .0005 .0005 .0005 .9985 mu = -2.34 Chance of winning any: 1/200 14990 100 200 -10 .005 .005 .005 .985 mu= 66.60 !! 2 dice 2 3 4 5 6 7 8 9 10 11 12 .027777 .055555 .083333 .111111 .1388888 .16666666 .1388888 .111111 .083333 .055555 .027777 3 dice 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 0.0046296 0.0138889 0.0277778 0.0462963 0.0694444 0.0972222 0.1157407 0.1250000 0.1250000 0.1157407 0.0972222 0.0694444 0.0462963 0.0277778 0.0138889 0.0046296 household size USA 1 2 3 4 5 6 7 .267 .336 .158 .137 .063 .024 .014 mu=2.518 var=2.00 sigma=1.415 unlicensed SW packages 0 1 2 3 4 .008 .076 .265 .412 .240 Cali Daily 4 0 1 2 3 4 .656 .292 .049 .004 0 right-skewed die 1 2 3 4 5 6 .5 .25 .125 .0625 .03125 .03125