Dylan asks for his lunch money every day from his mother. There is an 80% chance

Question

Dylan asks for his lunch money every day from his mother. There is an 80% chance she has money to give him and if so, he gets $5. However, if she does not have money Dylan has to give her $2. Whether Dylan gets $5 or gives out $2 is independent from day to day. Let x equal the number of times Dylan gets money during a 5-day week. Let w = the total amount he gets/gives for lunch in a week. (When he gives money just make the amount negative.)

a. As a function of x, what is w?
b. What is the expected value of x?  
c. What is the expected value of w?
d. What is the probability that x = 5?
e. What is the probability that x > 1?
f. What is the standard deviation of x?
g. What is the standard deviation of w?
h. What is the probability that w > 10?
i What is the probability that w is within 1 standard deviation of its expected value?

Explanation / Answer

We are allowed to do 4 subparts question at a time. Post again for more subparts of question.

a) Amount he gets/gives

w = 0.8 * x * 5 - 0.2 * (5 - x) * 2

w = 4x - 2 + 0.4x

w = 4.4x + 0.2

b) Expected value of x = 5 * 0.8 = 4

c) w = 4.4x + 0.2

w = 4.4 * 4 + 0.2

w = $17.8

d) x = 5

Every day it is an 80% chance

So, P = 0.8^5 = 0.33

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