Business Statistics CONTENTS Probability distribution functions - PowerPoint PPT Presentation

PROBABILITY DISTRIBUTIONS Business Statistics

CONTENTS Probability distribution functions (discrete) Characteristics of a discrete distribution Today we want to speed up. We will skip some slides or Example: uniform (discrete) distribution postpone a few. Prepare Example: Bernoulli distribution well, we want to start the Example: binomial distribution statistical topics as soon as possible. Probability density functions (continuous) Characteristics of a continuous distribution Example: uniform (continuous) distribution Example: normal (or Gaussian) distribution Example: standard normal distribution Back to the normal distribution Approximations to distributions Old exam question Further study

PROBABILITY DISTRIBUTION FUNCTIONS (DISCRETE) ▪ A sample space is called discrete when its elements can be counted ▪ We will code the elements of a discrete sample space 𝑇 as 1,2,3, … , 𝑜 or 0,1,2, … , 𝑜 − 1 ▪ Examples ▪ die 𝑦 ∈ 1,2,3,4,5,6 , so 𝑇 = 1,2,3,4,5,6 ▪ coin 𝑦 ∈ 0,1 ▪ number of broken TV sets 𝑦 ∈ 0,1,2, …

PROBABILITY DISTRIBUTION FUNCTIONS (DISCRETE) Distribution function 𝑄 𝑦 = 𝑄 𝑌 = 𝑦 ▪ the probability that the (discrete) random variable 𝑌 assumes the value 𝑦 ▪ alternative notation: 𝑄 𝑌 𝑦 Note our convention: capital letters ( 𝑌 ) for random variables lowercase letters ( 𝑦 ) for values

PROBABILITY DISTRIBUTION FUNCTIONS (DISCRETE) Example 1 if 𝑦 = 1 6 1 if 𝑦 = 2 6 1 if 𝑦 = 3 6 ▪ die: 𝑄 𝑦 = 1 if 𝑦 = 4 6 1 if 𝑦 = 5 6 1 if 𝑦 = 6 6 0 otherwise

PROBABILITY DISTRIBUTION FUNCTIONS (DISCRETE) Example: flipping a coin 3 times ▪ sample space 𝑇 = 𝐼𝐼𝐼, 𝐼𝐼𝑈, 𝐼𝑈𝐼, 𝑈𝐼𝐼, … ▪ define the random variable 𝑌 = number of heads 1 if 𝑦 = 0 8 3 if 𝑦 = 1 8 ▪ distribution function 𝑄 𝑦 = 3 if 𝑦 = 2 8 1 if 𝑦 = 3 8 0 otherwise 1 3 3 1 ▪ or: 𝑄 𝑌 0 = 8 , 𝑄 𝑌 1 = 8 , 𝑄 𝑌 2 = 8 , 𝑄 𝑌 3 = 8

PROBABILITY DISTRIBUTION FUNCTIONS (DISCRETE) ▪ 𝑄 𝑦 is a (discrete) probability distribution function (pdf or PDF) ▪ 𝑄 𝑦 = 𝑄 𝑌 = 𝑦 expresses the probability that 𝑌 = 𝑦 ▪ A random variable 𝑌 that is distributed with pdf 𝑄 is written as 𝑌~𝑄 ▪ Some properties of the pdf: ▪ 0 ≤ 𝑄 𝑦 ≤ 1 ▪ a probability is always between 0 and 1 ▪ σ 𝑦∈𝑇 𝑄 𝑦 = 1 ▪ the probabilities of all elementary outcomes add up to 1

PROBABILITY DISTRIBUTION FUNCTIONS (DISCRETE) ▪ A pdf may have one or more parameters to denote a collection of different but “similar”pdfs ▪ Example: a regular die with 𝑛 faces 1 𝑛 (for 𝑦 = 1, … , 𝑛 ) ▪ 𝑄 𝑌 = 𝑦; 𝑛 = 𝑄 𝑌 𝑦; 𝑛 = 𝑄 𝑦; 𝑛 = ▪ 𝑌~𝑄 𝑛 𝑛 = 4 𝑛 = 6 𝑛 = 8 𝑛 = 12 𝑛 = 20

PROBABILITY DISTRIBUTION FUNCTIONS (DISCRETE) In addition to the (discrete) probability distribution function (pdf) ▪ 𝑄 𝑌 = 𝑦 = 𝑄 𝑌 𝑦 = 𝑄 𝑦 we define the (discrete) cumulative distribution function (cdf or CDF) 𝐺 𝑦 = 𝐺 𝑌 𝑦 = 𝑄 𝑌 ≤ 𝑦 and therefore 𝑦 𝑦 𝐺 𝑦 = ෍ 𝑄 𝑌 = 𝑙 = ෍ 𝑄 𝑙 Depending on how we 𝑙=−∞ 𝑙=−∞ count, you may also start at 𝑙 = 0 or 𝑙 = 1

PROBABILITY DISTRIBUTION FUNCTIONS (DISCRETE) Example 1 ▪ die: 𝑄 𝑌 = 2 = 6 , but 𝑄 𝑌 ≤ 2 = 𝑄 𝑌 = 1 + 1 𝑄 𝑌 = 2 = 3 ▪ Some properties of the cdf: ▪ 𝐺 −∞ = 0 and 𝐺 ∞ = 1 ▪ monotonously increasing

PROBABILITY DISTRIBUTION FUNCTIONS (DISCRETE) ▪ pdf ▪ cdf

CHARACTERISTICS OF A DISCRETE DISTRIBUTION Expected value of 𝑌 𝑂 𝑂 𝐹 𝑌 = ෍ 𝑦 𝑗 𝑄 𝑌 = 𝑦 𝑗 = ෍ 𝑦 𝑗 𝑄 𝑦 𝑗 𝑗=1 𝑗=1 ▪ Example 1 die with 𝑄 1 = 𝑄 2 = ⋯ = 𝑄 6 = ▪ 6 1 1 1 1 1 1 7 1 ▪ 𝐹 𝑌 = 1 × 6 + 2 × 6 + 3 × 6 + 4 × 6 + 5 × 6 + 6 × 6 = 2 = 3 2 ▪ Interpretation: mean (average) ▪ alternative notation: 𝜈 or 𝜈 𝑌 so 𝐹 𝑌 = 𝜈 𝑌 ▪ ▪ Note difference between 𝜈 and the sample mean ҧ 𝑦 e.g., rolling a specific die 𝑜 = 100 times may return a mean ҧ 𝑦 = 3.72 or ▪ 3.43 while 𝜈 = 7/2 , always (property of die, property of “population”) ▪

CHARACTERISTICS OF A DISCRETE DISTRIBUTION Variance 𝑂 2 𝑄 𝑦 𝑗 var 𝑌 = ෍ 𝑦 𝑗 − 𝐹 𝑌 𝑗=1 ▪ Interpretation: dispersion alternative notation: 𝜏 2 or 𝜏 𝑌 2 or 𝑊 𝑌 ▪ so var 𝑌 = 𝜏 𝑌 2 ▪ ▪ Note difference between 𝜏 2 and the sample variance 𝑡 2 e.g., rolling a specific die 100 times may return a variance 𝑡 2 = 2.86 or 3.04 ▪ while 𝜏 2 = 35 12 , always (property of die, property of “population”) ▪ ▪ And of course: standard deviation 𝜏 𝑌 = var 𝑌

CHARACTERISTICS OF A DISCRETE DISTRIBUTION Transformation rules of random variable 𝑌 and 𝑍 ▪ For means: ▪ 𝐹 𝑙 + 𝑌 = 𝑙 + 𝐹 𝑌 ▪ 𝐹 𝑏𝑌 = 𝑏𝐹 𝑌 ▪ 𝐹 𝑌 + 𝑍 = 𝐹 𝑌 + 𝐹 𝑍 ▪ For variances: ▪ var 𝑙 + 𝑌 = var 𝑌 ▪ var 𝑏𝑌 = 𝑏 2 var 𝑌 ▪ if 𝑌 and 𝑍 independent (so if cov 𝑌, `𝑍 ): ▪ var 𝑌 + 𝑍 = var 𝑌 + var 𝑍 ▪ if 𝑌 and 𝑍 dependent: ▪ var 𝑌 + 𝑍 = var 𝑌 + 2cov 𝑌, 𝑍 + var 𝑍

EXAMPLE: UNIFORM DISTRIBUTION ▪ Generalization of fair die: ▪ equal probability of integer outcomes from 𝑏 through 𝑐 ▪ conditions: 𝑏, 𝑐 ∈ ℤ , 𝑏 < 𝑐 ▪ zero probability elsewhere ▪ uniform discrete distribution 1 𝑦 ∈ ℤ and 𝑦 ∈ 𝑏, 𝑐 ▪ pdf: 𝑄 𝑦; 𝑏, 𝑐 = ൝ 𝑐−𝑏+1 0 otherwise ▪ Examples: ▪ coin: 𝑏 = 0 , 𝑐 = 1 ▪ die: 𝑏 = 1 , 𝑐 = 6 ▪ Random variable: ▪ 𝑌~𝑉 𝑏, 𝑐

EXAMPLE: UNIFORM DISTRIBUTION No need to memorize or even discuss this sheet. Most information is either on the formula sheet or unimportant.

EXAMPLE: UNIFORM DISTRIBUTION ▪ Example: choose a random number from 1 through 100 with equal probability and denote it by 𝑌 ▪ random variable: 𝑌~𝑉 1,100 1 ▪ pdf: 𝑄 𝑦 = 𝑄 𝑌 = 𝑦 = 100 ( 𝑦 ∈ 1,2, … , 100 ) 𝑦 ▪ cdf: 𝐺 𝑦 = 𝑄 𝑌 ≤ 𝑦 = 100 ( 𝑦 ∈ 1,2, … , 100 ) 1 ▪ expected value: 𝐹 𝑌 = 50 2 9999 ▪ variance: var 𝑌 = 12 ≈ 833.25 ▪ Sample ( 𝑜 = 1000 ): ▪ values (e.g.): 45 , 96 , 33 , 7 , 44 , 96 , 20 , … ▪ mean: ҧ 𝑦 = 50.92 (e.g.) 2 = 823.25 (e.g.) ▪ variance: 𝑡 𝑦

EXERCISE 1 Given are two dice, with outcomes 𝑌 and 𝑍 . a. Find 𝐹 𝑌 + 𝑍 b. Find var 𝑌 + 𝑍

EXAMPLE: BERNOULLI DISTRIBUTION ▪ Bernoulli experiment ▪ random experiment with 2 discrete outcomes (coin type) ▪ head, true, “success”, female: 𝑌 = 1 ▪ tail, false, “fail”, male: 𝑌 = 0 ▪ Bernoulli distribution ▪ Examples: ▪ winning a price in a lottery (buying one ticket) ▪ your luggage arrives in time at a destination ▪ Probability of success is parameter 𝜌 (with 0 ≤ 𝜌 ≤ 1 ) ▪ 𝑄 1 = 𝑄 𝑌 = 1 = 𝜌 ▪ 𝑄 0 = 𝑄 𝑌 = 0 = 1 − 𝜌 ▪ Random variable ▪ 𝑌~𝐶𝑓𝑠𝑜𝑝𝑣𝑚𝑚𝑗 𝜌 or 𝑌~𝑏𝑚𝑢 𝜌

EXAMPLE: BERNOULLI DISTRIBUTION ▪ Expected value ▪ 𝐹 𝑌 = 𝜌 (obviously!) ▪ Variance ▪ var 𝑌 = 𝜌 1 − 𝜌 ▪ variance zero when 𝜌 = 0 or 𝜌 = 1 (obviously!) 1 ▪ variance maximal when 𝜌 = 1 − 𝜌 = 2 (obviously!) 𝜌 if 𝑦 = 1 ▪ pdf: 𝑞 𝑦; 𝜌 = ቐ 1 − 𝜌 if 𝑦 = 0 0 otherwise ▪ cdf: (not so interesting)

EXAMPLE: BINOMIAL DISTRIBUTION ▪ Repeating a Bernoulli experiment 𝑜 times ▪ 𝑌 is total number of “successes” ▪ 𝑄 𝑌 = 𝑦 is probality of 𝑦 “successes” in sample ▪ 𝑌 = 𝑌 1 + 𝑌 2 + ⋯ + 𝑌 𝑜 ▪ where 𝑌 𝑗 is the outcome of Bernoulli experiment number 𝑗 = 1,2, … , 𝑜 ▪ 𝑌 has a binomial distribution

EXAMPLE: BINOMIAL DISTRIBUTION ▪ Example ▪ flip a coin 10 times: 𝑌 is number of “heads up” ▪ roll 100 dice: 𝑌 is number of “sixes” ▪ produce 1000 TV sets: 𝑌 is number of broken sets ▪ What is important? ▪ the number of repitions ( 𝑜 ) ▪ the probability of success ( 𝜌 ) per item ▪ the constancy of 𝜌 ▪ the independence of the “experiments”

EXAMPLE: BINOMIAL DISTRIBUTION ▪ Expected value ▪ 𝐹 𝑌 = 𝑜𝜌 (obviously!) ▪ Variance ▪ var 𝑌 = 𝑜𝜌 1 − 𝜌 ▪ minimum ( 0 ) when 𝜌 = 0 or 𝜌 = 1 (obviously!) 1 ▪ maximum for given 𝑜 when 𝜌 = 1 − 𝜌 = 2 (obviously!) ▪ pdf: 𝑦! 𝑜−𝑦 ! 𝜌 𝑦 1 − 𝜌 𝑜−𝑦 𝑜! ( 𝑦 ∈ 0,1,2, … , 𝑜 ) ▪ 𝑞 𝑦; 𝑜, 𝜌 = ▪ cdf: Recall the factorial function: 𝑦 ▪ 𝐺 𝑦; 𝑜, 𝜌 = σ 𝑙=0 𝑞 𝑦; 𝑜, 𝜌 5! = 5 × 4 × 3 × 2 × 1 ▪ Random variable: ▪ 𝑌~𝑐𝑗𝑜 𝑜, 𝜌 or 𝑌~𝑐𝑗𝑜𝑝𝑛 𝑜, 𝜌