Stats 1430 Chapter 6

Types of Random Variables

-Discrete Random Variable - Continuous Random Variable

Discrete Random Variable

-Finite: X= 0,1,2,3 -Countably infinite: X=1,2,3...... Example: -# of yeses on 10 calls -# of calls until a yes

What the difference between Discrete Random Variables and Continuous Random Variables mean to us? (Sums v. Integrals)

-The way you define the distribution is different for each type -The way you describe and handle probabilities is different for each type -The way you calculate the mean and standard deviation is different for each type

Continuous Random Variable

-Uncountably Infinite: X= {[0,1] or (0,infinity) Example: -Measurements, times, scores, etc

Example: House Values 1) Mean value of a home with a mortgage in 2005: $187,000 2) Mean value of a home without a mortgage in 2005: $127,100

1) μx 2)μy Mean difference in home values: μ(x-y) = μx - μy = 187,000 - 127,100 = 59,900

The standard deviation of a discrete random variable

Def: A weighted average of the deviation from the mean -Square root of variance Notation: σx Formula: square-root(∑(x - μx)^2 * P(x)) Units: Same as X

The mean of a discrete random variable

Def: A weighted average of the possible outcomes; weights are the probabilities -Average of all possible values in whole population and not the sample. - Xbar cannot equal μx -Possible outcomes: X1, X2,...Xk -Probabilities: P1, P2,...Pk Notation: μx = mean of all X values in the population Formula (Value Probability): μx = ∑X P(x) = Value * Probability Summed

The variance of a discrete random variable

Def: A weighted average of the squared deviations from the mean -Measure of variability square of SD Notation: σx^2 (sigma) Formula: ∑(x - μx)^2 * P(x) Units? No units

Random Variables

Def: Characteristic you measure, count, or categorize. Randomly changes with each individual Notation: X, Y, Z Examples: - X = number of heads on 2 coin flips - X = number of customers in a queue at the bank - X = time it takes to serve a customer at the help desk - X = gender of customers (Category)

Probability Distribution of a discrete random variable

Def: List of all possible values and probability is how often we expect each to occur Notation: P(x) = Probability the X (for all x) takes on the value X Two Requirements: 1) 0 <= P(x) <= 1 2) ∑P(x) = 1 (For all of x)

Rules of Means # 1 - Linear Transformations

Def: Multiply X values by "b" and add a # called "a" y = a + bx Example: 10% raise + 500 y = 500 + 1.10x μy = μ500 + 1.10x Rule: μa +bx = a + bμx = 500 + 1.10μx *This rule holds for any X or Y

The mean temperature in Cancun is 28.5 degrees Centigrade (C). What is the mean temperature in Fahrenheit (F)?

F = (9/5)C + 32 -Example: 0 degrees C = 32 degrees F μf = μ(9/5)C + 32 = 32 + (9/5)μc =32 + (9/5)(28.5) = 83.3 degrees F

Discrete Example: Family Size (k) # Family Prob 1---------0 2--------0.42 3--------0.23 4--------0.21 5--------0.09 6--------0.03 7--------0.02

Mean Family Size: μx = (2*0.42) + (3*0.23) + (4*0.21) + (5*0.09) + (6*0.03) + (7*0.02) = 3.19 Interpret this result: Avg. family size expected to be 3.14 people

Rule #2 of Means - Sums and Differences

Rule: 2 random variables X + Y μ(x+y) = μx + μy Example: - Mean of X = 10 and mean of Y = 20 - What is the mean of X+Y? μ(x+y) = μx + μy = 10 + 20 = 30 -What is the mean of X-Y? μ(x-y) = μx + μ-1y = μx +- μy = μx - μy = 10 - 20 =-10 Note: This rule holds for any X, Y

μx = 10 μy = 30 What is... μ(2x+3y) μ(2x-3y)

μ(2x+3y) = μ2x + μ3y = 2μx + 3μy = 2(10) + 3(30) = 110 μ(2x-3y) = μ2x - μ3y = 2μx - 3μy = 2(10) - 3(30) = -70