Discrete Random Variables and Their Probability Distributions

Introduction

• The first few lectures come from Mathematical Statistics with Applications, by Wackerly.
  • We must understand the underlying probability and random variable theory before moving into the Bayesian world.
• We will be covering the following chapters:
  • Chapter 2: probability theory
  • Chapter 3: discrete random variables
  • Chapter 4: continuous random variables

3.1: Basic Definitions

• Discrete random variable: a variable that can assume only a finite or countably infinite number of distinct values.

• Probability distribution of a random variable: collection of probabilities for each value of the random variable.

• Notation:

  • Uppercase letter (e.g., Y) denotes a random variable.
  • Lowercase letter (e.g., y) denotes a particular value that the random variable may assume.
    • The specific observed value, y, is not random.

3.2: Probability Distributions for Discrete RV

• probability function for Y: sum of the the probabilities of all sample points in S that are assigned the value y

  • P[Y = y] = p(y): the probability that Y takes on the value y.

• probability distribution for Y: a formula, table, or graph that provides p(y) = P[Y = y] \forall y.

• Theorem:

  • For any discrete probability distribution, the following must be true:
    • 0 \le p(y) \le 1 \ \forall \ y
    • \sum_y p(y) = 1 \ \forall \ p(y) > 0.

3.2: Probability Distributions for Discrete RV

• A supervisor in a manufacturing plant has three men and three women working for them. The supervisor wants to choose two workers for a special job. Not wishing to show any biases in their selection, they decides to select the two workers at random.

• Let Y denote the number of women in his selection. Find the probability distribution for Y.

3.2: Probability Distributions for Discrete RV

• When the health department tested private wells in a county for two impurities commonly found in drinking water, it found that:
  • 20% of the wells had neither impurity,
  • 40% had impurity A, and
  • 50% had impurity B.
• If a well is randomly chosen from those in the county, find the probability distribution for Y, the number of impurities found in the well.
  • Hint: some wells had both impurities…

3.3: Expected Values

• Expected value: Let Y be a discrete random variable with the probability function, p(y). Then, the expected value of Y, E[Y], is defined to be

E(Y) = \sum_{y} y p(y)

• When p(y) is an accurate characterization of the population frequency distribution, then the expected value is the population mean.

E[Y] = \mu

• Theorem:

  • Let Y be a discrete random variable with probability function p(y) and g(Y) be a real-valued function of Y (i.e., a transformed variable). Then the expected value of g(Y) is given by

E[g(Y)] = \sum_{y} g(y) p(y)

3.3: Expected Values

• Variance: if Y is a random variable with mean E[Y] = \mu, the variance of a random variable Y is defined to be the expected value of (Y-\mu)^2.

V[Y] = E\left[ (Y-\mu)^2 \right]

• If p(y) is an accurate characterization of the population frequency distribution, then V(Y) is the population variance,

V[Y] = \sigma^2

• Standard deviation: the positive square root of V[Y].

3.3: Expected Values

• The probability distribution for a random variable Y is given below.

• Find the mean, variance, and standard deviation of Y.

3.3: Expected Values

• Theorem:

  • Let Y be a discrete random variable with probability function p(y) and c be a constant. Then,

E(c) = c

• Theorem:

  • Let Y be a discrete random variable with probability function p(y), g(Y) be a function of Y, and c be a constant. Then,

E[cg(Y)] = cE[g(Y)]

• Theorem:

  • Let Y be a discrete random variable with probability function p(y), and g_1(Y), g_2(Y), ..., g_k(Y) be k functions of Y. Then,

E[g_1(Y) + g_2(Y) + ... + g_k(Y)] = E[g_1(Y)] + E[g_2(Y)] + ... + E[g_k(Y)]

3.3: Expected Values

• Putting the previous theorems into one:

• Theorem:

  • Let Y be a discrete random variable with probability function p(y) and mean E[Y] = \mu. Then,

V[Y] = \sigma^2 = E\left[(Y-\mu)^2\right] = E\left[Y^2\right] - \mu^2

3.3: Expected Values

• The probability distribution for a random variable Y is given below.

• Use the previous theorem to find V[Y] and compare to our previous answer.

  • Recall that \mu=1.75.

3.3: Expected Values

• Let Y be a random variable with p(y) in the table below.

• Find
  • E[Y]
    
    
  • E[1/Y]
    
    
  • E\left[Y^2-1\right]
    
    
  • V[Y]

3.3: Expected Values

• E[Y]
  
  
  
  
  
• E[1/Y]
  
  
  
  
  
• E\left[Y^2-1\right]
  
  
  
  
  
• V[Y]

3.4: Binomial Probability Distribution

• Binomial experiment:

  1. The experiment consists of a fixed number, n, of identical trials.

  2. Each trial results in one of two outcomes: success (S) or failure (F).

  3. The probability of success on a single trial is equal to some value p and remains the same from trial to trial.

     • The probability of failure is equal to q = (1-p).

  4. The trials are independent.

  5. The random variable of interest is Y, the number of successes observed during the n trials.

3.4: Binomial Probability Distribution

• Binomial Distribution

  • A random variable Y is said to have a binomial distribution based on n trials with success probability p iff

p(y) = {n \choose y}p^y q^{n-y}, \text{ where } y = 0, 1, 2, ..., n, \text{ and } 0 \le p \le1

• Theorem:

  • Let Y be a binomial random variable based on n trials and success probability p. Then

E[Y] = \mu = np \ \ \ \text{and} \ \ \ V[Y] = \sigma^2 = npq

• See Wackerly pg. 107 for derivation.

3.4: Binomial Probability Distribution

p=0.10

3.4: Binomial Probability Distribution

p=0.25

3.4: Binomial Probability Distribution

p=0.50

3.4: Binomial Probability Distribution

p=0.75

3.4: Binomial Probability Distribution

p=0.90

3.4: Binomial Probability Distribution

• What do you notice when comparing distributions under p vs. 1-p?

3.4: Binomial Probability Distribution

• What do you notice as n increases?

3.4: Binomial Probability Distribution

• The manufacturer of a low-calorie dairy drink wishes to compare the taste appeal of a new formula (formula B) with that of the standard formula (formula A). Each of four judges is given three glasses in random order, two containing formula A and the other containing formula B. Each judge is asked to state which glass he or she most enjoyed. Suppose that the two formulas are equally attractive. Let Y be the number of judges stating a preference for the new formula.

  1. Find the probability function for Y.
     
     
     

  2. What is the probability that at least three of the four judges state a preference for the new formula?
     
     
     

  3. Find the expected value of Y.
     
     
     

  4. Find the variance of Y.

3.8: Poisson Probability Distribution

• The Poisson probability distribution often provides a good model for the probability distribution of the number Y of rare events that occur in space, time, volume, or any other dimension.

• Poisson Distribution:

  • A random variable Y is said to have a Poisson probability distribution iff

p(y) = \frac{\lambda^y}{y!}e^{-\lambda}, \text{ where } y=0,1,2,..., \text{ and } \lambda > 0

• Theorem

  • If Y is a random variable with a Poisson distribution with parameter \lambda, then

E[Y] = \mu = \lambda \text{ and } V[Y] = \sigma^2 = \lambda

• See Wackerly pg. 134 for derivation.

3.8: Poisson Probability Distribution

• Customers arrive at a checkout counter in a department store according to a Poisson distribution at an average of seven per hour. During a given hour, what are the probabilities that

  1. no more than three customers arrive?
     
     
     
     
     

  2. at least two customers arrive?
     
     
     
     
     

  3. exactly five customers arrive?
     
     
     
     
     

Homework

• 3.6
• 3.10
• 3.15
• 3.22
• 3.34
• 3.60
• 3.128
• 3.129
• 3.136