Discrete probability distributions
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Discrete Probability Distributions. Random Variables. Expected Value (mean, average). Μ = E(X) = Σ value(x) x probability(x). Expected value = 0.5 x (-1) + 0.5 x (+1) = 0. Example: Lottery of 1,000 tickets, with the following payout structure, has an E(x) = $1.00.

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Expected value mean average
Expected Value (mean, average)

Μ = E(X) = Σ value(x) x probability(x)




Variance s 2 or spread and standard deviation s
Variance s structure, has an E(x) = $1.00.2 (or “spread”) and Standard Deviation s


Calculating the Variance s structure, has an E(x) = $1.00.2 and the Standard Deviation s.

Var(Y) = s2 = Σ[(y – E(Y))2 x P(y)]

Or Var(Y) = s2 = Σ[P(y) x Y] – E(Y)2

Sdev s = (s2)0.5


Mathematics
Mathematics structure, has an E(x) = $1.00.

Factorials !

n! = n x (n – 1) x (n – 2) …. x 1

Example: 6! = 6 x 5 x 4 x 3 x 2 x 1 = 720.


Binomial distribution
Binomial Distribution structure, has an E(x) = $1.00.

  • n Bernoulli trials, where each trial only has two possible outcomes, like a coin toss (heads or tails) or baby (boy or girl).

  • p = the probability of success in each trial, and (1 – p) is the probability of failure.

  • The probability of x success in n trials is ……


Example: a die is rolled exactly n = 5 times. What is the probability of rolling exactly x = 2 sixes? (Note the probability of rolling a 6 is P(six) = 1/6 = 0.166667.)


Binomial calculators online
Binomial Calculators (online) probability of rolling exactly x = 2 sixes? (Note the probability of rolling a 6 is P(six) = 1/6 = 0.166667.)

http://stattrek.com/tables/binomial.aspx

Or, using MS Excel, go to Formulas/More Functions/Statistical/BINOMDIST


Cumulative binomial p robabilities
Cumulative Binomial probability of rolling exactly x = 2 sixes? (Note the probability of rolling a 6 is P(six) = 1/6 = 0.166667.)Probabilities

  • As before, a question can be about the probability of exactly x successes in n trials. P(X = x).

  • But the question can also be about the (Cumulative) probability of getting x or less successes. P(X ≤ x).

  • Example: P(X ≤ 3) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3).

  • The probability of getting at least 4 successes in n trials = 1 – P(X ≤ 3).


Binomial distribution statistics
Binomial Distribution Statistics probability of rolling exactly x = 2 sixes? (Note the probability of rolling a 6 is P(six) = 1/6 = 0.166667.)

Mean = np

Variance = np(1 – p)


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