On Valentine\'s Day the Quiet Nook restaurant offers a Lucky Lovers Special that

Question

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 also brings a scratch-off ticket that the couple scrapes to reveal a hidden number; call this number X. The value of the number X is the dollar amount that will be subtracted from their dinner bill. The possible values of the scratch-off number X are shown in the table below. The table also shows the probability distribution of the values of X.

Question 1. Find the expected value of the discount X.
$

Question 2. Find the standard deviation of the discount X.(Use 2 decimal places.)
$

For the past several weeks the restaurant has also been distributing coupons worth $5 off any dinner for two. The coupon can be used in addition to the discount X from the scratch-off ticket. If every couple dining there on Valentine's Day also brings a $5 coupon, what are the expected value and standard deviation of the total discount the couples receive?

Question 3. Expected value of total discount.
$

Question 4. Standard deviation of total discount
$

When two couples dine together on a single check, the restaurant doubles the scratch-off discount X; so the discount values are 2X, that is, twice the amounts shown in the above table. (the $5 discount coupon cannot be used).

Question 5. What is the expected value of the discount for 2 couples?
$

Question 6. What is the standard deviation of the discount for 2 couples?
$

Discount Value X 2 10 19 26 31 Probability 0.25 0.30 0.25 0.10 0.10

Explanation / Answer

1)

expected value of discount E(X) =13.95

2) std deviation =9.51

3) as now new discount Y =X+5

hence expected value =E(Y) =E(X)+5 =13.95+5=18.95

4) std deviation SD(Y)=SD(X) = 9.51

5)

here as Y =2X

hence expected Value E(Y) =2*E(X) =2*13.95 =27.90

6) std deviation SD(Y) =(22Var(X))1/2 =2*SD(X) =2*9.51 =19.01

x p(x) xP(x) (x-)2 (x-)2P(x) 2 0.250 0.500 142.803 35.701 10 0.300 3.000 15.603 4.681 19 0.250 4.750 25.503 6.376 26 0.100 2.600 145.203 14.520 31 0.100 3.100 290.703 29.070 total 1 = 13.95 619.813 2= 90.3475 std deviation=     =    2 = 9.5051

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