Chapter 16 Random Variables Random Variable Variable that assumes any of several different values as a result of some random event. Denoted by X Discrete (finite number of outcomes) Continuous Probability Model or Probability Distribution Collection of all the possible values and their corresponding probabilities

Random Variables Ex: Insurance Company Charges $50 per policy Death $10000 Disability $5000 Policy holder outcome Payout X Probability P(X=x) Death10,0001/1000 Disability5,0002/1000 Neither0997/1000

Random Variables Ex: Insurance Company Expected Value of a Random Variable (or Mean) E(X) = 10,000(1/1,000)+5000(2/1000)+0(997/1000)

Random Variables Standard Deviation of the random variable Exercises page 427, just checking 411 Policy holder outcome Payout X Probability P(X=x) Deviation Death10,0001/1000(10,000-20) Disability5,0002/1000(5,000-20) Neither0997/1000(0-20)

More About Means and Variances Shift Adding or subtracting a constant from the data shifts the expected value, but doesn’t change the variance or standard deviation E(X±c) = E(X) ± c Var(X±c) = Var(X) Rescaling Multiplying each value of a random variable by a constant multiplies the expected value and the standard deviation by that constant E(aX)= a E(X) SD(aX) = a SD(X) Var(aX) = a 2 Var(X)

More About Means and Variances Adding or subtracting random variables The mean of the sum of two random variables is the sum of the means E(X+Y) = E(X) + E(Y) The mean of the difference of two random variables is the difference of the means E(X-Y) = E(X) - E(Y) If the random variables are independent, the variance of their sum or difference is always the sum of the variances Var(X±Y) = Var(X) +Var(Y) Just checking 418

Continuous Random Variables We don’t have discrete outcomes, the random variable can take on “any” value. Example Packaging Stereos PackingNormal E(P)=9SD(P)=1.5 BoxingNormal E(B)=6SD(B)=1 What is the probability that packing two consecutive systems takes over 20 minutes? What percentage of the stereo systems take longer to pack than to box?

Chapter 17 Probability Models Bernoulli Trials Only two possible outcomes Success Failure The probability of a success denoted “p” is the same on every trial. The trials are independent If this assumption is violated, it is still ok to proceed as long as the sample is smaller than 10% of the population. Ex: Coin toss, rolling a die ?

Geometric probability model for Bernoulli trials: Geom (p) p = probability of success q =1-p probability of failure X = number of trials until the first success occurs P(X=x)=q x-1 p Expected Value Standard deviation

The Binomial Model Ex: p=1/6 q=5/6 Probability of 2 successes in 5 trials? (5/6) 3 (1/6) 2 What about the other orders? The number of different orders in which we can have k successes in n trials is written n C k and pronounced “n choose k” (The C actually stands for combinations) Where n!=1 x 2 x 3 x … x (n-1) x n

Binomial probability model for Bernoulli trials: Binom (n,p) n = number of trials p = probability of success q = 1 – p probability of failure K = number of successes in n trials P(K=k)= n C k p k q n-k where Mean Standard deviation

Exercises Step-by-step page 435 and 438