Discrete probability distributions, stats, histogram; expected value

A discrete probability distribution consists of all the values a random variable can assume and the corresponding probabilities of the values.
The probabilities are determined theoretically or by observation, i.e. they are all known. They sum to 1.

RV X values: in increasing order, please
P(x) values:
    

n=
Mean or expected value E[X]= μ = ∑(x·P(x)) :
   Expectation: what the result of a probability experiment would be on average, over the long term.
Variance σ2 = ∑((x-μ)2·P(x))= ∑(x2·P(x))-μ2:    = E[(X-μ)2]= E(X2)-μ2
Standard deviation σ =

Cumulative probability:

E(X2) = ∑(x2·P(x))    = σ2+μ2

Probability histogram: 

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

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

Lottery/raffle:
Expected value of ticket= (payoff-price)*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: $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 

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

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