Discrete probability distributions, stats, histogram; expected value
A discrete probability distribution consists of all the values or outcomes a random variable
can assume and the corresponding probabilities of the values.
(Discrete means a finite number of values.
[countable].)
The probabilities are determined theoretically or by observation, i.e. they are all known.
They sum to 1.
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
1 die
1 2 3 4 5 6
.16666 .16666 .16666 .16666 .16666 .16666
2 dice sum
2 3 4 5 6 7 8 9 10 11 12
.027777 .055555 .083333 .111111 .1388888 .16666666 .1388888 .111111 .083333 .055555 .027777
2 dice product
X: 1 2 3 4 5 6 8 9 10 12 15 18 20 24 25 30 36
freq: 1 2 3 3 2 4 2 1 2 4 2 2 2 2 1 2 1
P(x): .02777 .05555 .08333 .08333 .05555 .11111 .05555 .02777 .05555 .11111 .05555 .05555 .05555 .05555 .02777 .05555 .02777
3 dice sum
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
4 coins, #H
0 1 2 3 4
.0625 .25 .375 .25 .0625
4 coins, #H-#T
0 2 -2 4 -4
.375 .25 .25 .0625 .0625
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
payoff - price = winnings (net gain, profit)
Lottery/raffle:
Expected value of ticket= (payoff-price)*chance_of_winning + -price*(1-chance_of_winning)
= winnings*chance_of_winning + -price*(1-chance_of_winning)
x P(x)
----- -----------
payoff-price prob. of winning
(net gain)
-price prob. of losing = 1-prob.win
Price: $1 Payoff: $100 #tickets:200 --> chance of winning: 1/200 = .005
99 -1
.005 .995
E[ticket]=-$.495
Price: $10 Payoff: $15000 #tickets:2000 --> chance of winning: 1/2000
14990 -10
.0005 .9995
E[ticket]= -$2.50
Pick 3 $1 ticket, $500 payout, 1/1000 chance of winning
499 -1
.001 .999
E[ticket]= -$.50
Cali Daily 4 ??
0 1 2 3 4
.656 .292 .049 .004 0
Roulette. $1 bet. one number: 1/38 chance, $36 payoff
35 -1
0.026315789 0.97368421
1/38 37/38
Roulette. $1 bet. one color (red or black): 18/38 chance, $2 payoff
1 -1
0.47368421 0.526315789
18/38 20/38
American roulette has 22 bets. 21 have the same expected value.
Coin flip: heads I win, tails you win. $1 bet
1 -1
.5 .5
E(X)=0 is a fair game.
Coin flip: heads I win, tails you lose. 😉 $1 bet
1 -1
1 0
Rock-Paper-Scissors
Win Lose Tie
1 -1 0
.33333 .33333 .33333
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 !!
30% chance profit 100, 40% chance profit 200, 30% chance profit -300
Irregular / Empirical Discrete Distributions
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
pets per household
0 1 2 3 4
.40 .30 .18 .09 .04
right-skewed die
1 2 3 4 5 6
.5 .25 .125 .0625 .03125 .03125
sum 2 right-skewed dice
2 3 4 5 6 7 8 9 10 11 12
1/4 1/4 3/16 1/8 5/64 1/16 7/256 3/256 5/1024 1/512 1/1024
.25 .25 .1875 .125 .078125 .0625 .02734375 .01171875 .0048828125 .001953125 .0009765625
Number of children born per woman USA
0 1 2 3 4 5
.2 .18 .35 .17 .07 .03
19th hole at Duffers' Tournament of Players. Golf
-3 -2 -1 0 1 2 3 eagle, birdie, par, bogey, double bogey,
.01 .03 .1 .26 .30 .25 .05
Prescription dosage for Med X
2.5 5.0 7.5 10.0
.45 .30 .15 .10