There are two "independent" games: • in the first game, the possible rewards are {0, 1, 2} and probabilities of getting the rewards are: p(0) = p(1) = p(2) = 1 3 . • in the second game, the possible rewards are {10, 20, 30} and probabilities of getting the rewards are: p(10) = 1 2 , p(20) = 1 3 , p(30) = 1 6 . define x to be the multiplication of the rewards obtained from game 1 and game 2. part a calculate pmf of x, e[x] and var[x]. part b calculate e[2x + 4] and var[3x + 6].

x = central angle apc = measure of minor arc ac

the two angles x and 30 add to 180, so x+30 = 180 and x = 180-30 = 150. the measure of minor arc ac is 150 degrees.

what we need to do from here is find the circumference and then multiply it by x/360 since there are 360 degrees in a circle. in this case, x = 150, so x/360 = 150/360 = 15/36 = 5/12. the shortest distance along the curve from a to c is 5/12 of the distance around the entire circle (aka circumference)

c = circumference

c = pi*d

c = pi*24

c = 24pi

now multiply this by 5/12 to get

distance along minor arc ac = (5/12)*(circumference)

distance along minor arc ac = (5/12)*(24pi)

distance along minor arc ac = 10pi

the formula you can use is

l = (x/360)*pi*d

with

l = length of minor arc

x = central angle

d = diameter

which essentially is the steps we took though a bit differently.