Random Variables “a random phenomenon (event) whose numerical outcome is unknown”

Discrete Random Variables Countable number of possible values Value of x Probability Each value of x has its probability In a legitimate probability distribution, probabilities add to 100% SWITCHES

Quiet Nook Restaurant On Valentine’s Day the Quiet Nook restaurant offers a Lucky Lovers Special that could save couples money on their romantic dinners. When the waiter brings the check, he’ll also bring the four aces from a deck of cards. He’ll shuffle them & lay them face down on the table. The couple chooses 1 card to turn over. If it’s a black ace, they pay the full amount of the dinner. But, if it’s the ace of hearts, the waiter will give them a $20 Lucky Lovers discount. If they first turn over the ace of diamonds (red!), they turn over one of the remaining cards– if it’s a heart, they earn a $10 discount.

Expected Value Standard Deviation Notation E(x) or µ x σ x Formula The mean µ x is the same as the expected value E(X) The standard deviation σ x is the square root of the variance σ x 2

At the Quiet Nook: Ace Hearts = -$20; Ace Spades = -$0; Ace Diamonds & then heart = -$10 What is the expected discount for a couple? Let x = discount P(X = 20) = P(H) = ¼ P(X = 10) = P(D, then H) = P(D) P(H D) = (¼) (⅓ ) = 1/12 P(X = 0) = P(X ≠ 20 or 10) = 1 – (1/4 + 1/12 ) = 2/3 OutcomeAce of H Ace D, then H Black Ace X P(X)1/41/122/3 E(X) = 20(1/4) + 10(1/12) + 0(2/3) = 5.83 Couples can expect an average discount of $5.83.

At the Nook… OutcomeAce of H Ace D, then H Black Ace X P(X)1/41/122/3 What is the standard deviation of the discount? E(X) = $5.83 Find the variance: Couples can expect the discount to average $5.83, with a standard deviation of $8.62

Still at The Nook …. Suppose for several weeks the restaurant has also been distributing coupons worth $5 off any one meal (one discount per table). If every couple dining there on Valentine’s Day brings a coupon, what will be the mean & standard deviation of the total discounts they’ll receive? E(X) = $5.83 and = $8.62 Let D = total discountD = X + 5 E(D) = E(X + 5) = E(X) + 5 = $10.83 Var (D) = Var ( X + 5) = Var (X) = $ Couples with the coupon can expect total discounts of $10.83 with the standard deviation of still $8.62.

Expected Values Standard Deviations Notation E(x) or µ x σ x Formula a · X a·E(x) X ± a E(x) ± a σ x

Check please… If two couples dine together, is the bill any different if they use a single check or ask for separate checks? Single check: E(2X) = 2E(X) = 2(5.83) = $11.66 To find the Standard Deviation: Var(2X) = 2 2 Var(X) = 2 2 ( ) = Standard Dev for single check = $17.24 Separate Checks: E(X+Y) = E(X) + E(Y) = = $11.66 Var(X+Y) = Var(X) + Var(Y) = Standard Dev for separate checks = $12.19 The mean would be the same, but the standard deviation would be much smaller for separate checks. It makes a difference!

At a little shop around the corner… A competing restaurant has a Lottery of Love discount– inside a wrapped chocolate. The manager says the discounts vary with a mean of $10.00 and a standard deviation of $15.00 How much more can you expect to save at this restaurant than at the Nook ? Let W = this restaurant’s discount and X= Quiet Nook’s discount E(W - X) = E(W) – E(X) = – 5.83 = $4.17 SD(W-X) = =$17.30 The LOL discount will average $4.17 higher with a standard deviation of $17.30

Continuous Random Variables with a Uniform Distribution The graph of the probability distribution will be a rectangle whose height = 1/length The area under this density curve = 1 Find the distance of the interval & multiply by the height Outside the rectangle, the probability = 0

Continuous Random Variables with a Normal Distribution Standardize the values by finding the z score: Use Normalcdf (low z, high z) to find the probability of the interval

Cautions for 2 Random Variables Be sure variables are independent Don’t assume everything has a Normal Distribution Means (expected values) add IF the variables are independent. Variances add, standard deviations do not! Variances of independent random variables add even when finding the differences between two random variables