Chapter 5: Discrete Probability Distributions

MAT 205 @ Hampton University

random variable

a variable whose numeric value is determined by the outcome of a probability experiment

Probability Distribution

a table or formula that gives the probability for every value of the random variable X

• 0 ≤ P (X = ⅹ) ≤ 1

  • X: Concept/Event

  • ⅹ: placeholder for the value

Expected Value

for discrete variable X is also known as the mean of the probability of X

Formula:

E (X) = μ = ∑ [ⅹ_i ∙ P (X = ⅹ_i)]

variance for a probability distribution

Formula

σ² = E (X²) - [ E(X)]²

• E (X²): ∑ ⅹ²P(X = ⅹ)

• [ E(X)]²: ∑ ⅹP(X = ⅹ)

standard deviation for a probability distribution

σ = √σ² = √E (X²) - [ E(X)]²

Bernoulli Trial (Binomial Distribution)- Properties

1. The experiment has a fixed number, “n,“ identical trials

2. Each trial is independent of the others

3. Only two possible outcomes for each trial, being success or failure

4. The probability of success is p, the probability of failure is q = 1 - p

5. The binomial random variable X, counts the number of successes across n trials

Bernoulli Trial (Binomial Distribution)- Formula

P(X = ⅹ) = _nC_ⅹ ∙ p^ⅹ ∙ (1 - p)^{(n - ⅹ)}

• n: # of times trial is repeated

• ⅹ: # of wanted successes

• p: probability of success

Bernoulli Trial (Binomial Distribution)- Mean (Expected Value) & Variance

• µ = np

• σ² = np(1 - p)

  • = npq

Poisson Experiment (Phenomenon)- Properties

1. Each success must be independent of any other successes

2. The Poisson random variable, X, counts the number of successes in a given interval

3. The mean number of success in a given interval must remain constant

Poisson Experiment (Phenomenon)- Formula

P(X = ⅹ) = (e^-λ ∙ λ^ⅹ) / ⅹ!

• e: specific # like π is

• λ: mean # of successes in each interval

• ⅹ: # of wanted successes

Poisson Experiment (Phenomenon)- Mean (Expected Value) & Variance

µ = σ² = λ

Hypergeometric Distribution- Properties

1. Each trial consists of selecting 1 of the N items in the population and results in either a success or a failure

   1. This means the population size is known

2. The experiment consists of n trials

3. The total number of possible successes is K

4. The trials are dependent

5. The Hypergeometric random variable, X, counts the number of successes in n trials

Hypergeometric Distribution- Formula

P(X = ⅹ) = [ ( _KC_ⅹ )( _{N - K}C_{n - ⅹ} ) ] / ( _N C_n )

• ( _KC_ⅹ ): combination population successes, CHOOSE sample successes

• ( _{N - K}C_{n - ⅹ} ): combination population failures, CHOOSE sample failures

• ( _N C_n ): combination population size, CHOOSE sample size

Hypergeometric Distribution- Mean

E (X) = µ = n ∙ [ K / N ]

Hypergeometric Distribution- Variance

σ² = [ ( N - n ) / ( N - 1) ] ∙ n ∙ ( K / N ) ∙ [ ( 1 - K / N ) ] or [ nK ( N - K ) (N - n) ] / [ N² ( N - 1 ) ]