Chapter 16 Random Variables

1. Chapter 16Random 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

2. Random Variables • Ex: Insurance Company • Charges $50 per policy • Death $10000 • Disability $5000

3. 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)

4. Random Variables • Standard Deviation of the random variable • Exercises page 427, just checking 411

5. 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) = a2 Var(X)

6. 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

7. Continuous Random Variables • We don’t have discrete outcomes, the random variable can take on “any” value. • Example • Packaging Stereos • Packing Normal E(P)=9 SD(P)=1.5 • Boxing Normal E(B)=6 SD(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?

8. Chapter 17Probability 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 ?

9. 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)=qx-1p • Expected Value • Standard deviation

10. 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 nCk and pronounced “n choose k” (The C actually stands for combinations) • Where n!=1 x 2 x 3 x … x (n-1) x n

11. 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)= nCk pkqn-k • where • Mean • Standard deviation

12. Exercises • Step-by-step page 435 and 438