Question

Two best friends sell cupcakes. They have 3 boxes red velvet , 5 boxes vanilla, 4 boxes chocolate, and 6 peanut butter. Each box they sells for $4.

a)   Define random variable
b)   Determine probability distribution and parameters for the random variable defined in item a.
c)   Suppose that after two hours ten boxes of cupcakes have been purchased. Determine the cumulative distribution function for the number of red velvet purchased
d)   Draw the probability distribution function for the number of red velvet purchased.

Answer 1

a.

random variable X : type of cupcake they sell

it can be : { red velvet , vanilla, chocolate, peanut butter }

b.

parameter :

total boxes = 3+5+4+6 = 18

distribution :

P(X) = no. of boxes of X / total boxes

P(red velvet) = 3/18 = 1/6 = 0.1667

P(vanilla) = 5/18 = 0.2777

P(chocolate) = 4/18 = 0.2222

P(peanut butter) = 6/18 = 1/3 = 0.3333

c.

x = no. of red velvet purchased out of 10

x can be : 0,1,2,3

no. of red velvet boxes = 3

no. of non red velvet boxes = 18-3 = 15

P(x red velvet purchased out of 10) = (select x red velvet out of 3)*(select (10-x) non red velvet out of 15) / (select 10 boxes from 18 boxes)

= 3Cx * 15C(10-x) / (18C10)

= 3Cx * 15C(10-x) / 43758

cummulative probability distribution = P(x<=X) = P(0) + P(1) + ...+P(X)

X	P(X)	P(x<=X)
0	3C0* 15C(10-0) / 43758 = 0.0686	0.0686
1	3C1 * 15C(10-1) / 43758 = 0.3431	0.0686+0.3431= 0.4117
2	3C2 * 15C(10-2) / 43758 = 0.4412	0.4412+0.0686+0.3431 = 0.8529
3	3C3 * 15C(10-3) / 43758 = 0.1471	0.1471+0.4412+0.0686+0.3431 = 1

d.

probability distribution function for the number of red velvet purchased :

P(x red velvet purchased out of 10) = (select x red velvet out of 3)*(select (10-x) non red velvet out of 15) / (select 10 boxes from 18 boxes)

= 3Cx * 15C(10-x) / (18C10)

= 3Cx * 15C(10-x) / 43758

X	P(X)
0	3C0* 15C(10-0) / 43758 = 0.0686
1	3C1 * 15C(10-1) / 43758 = 0.3431
2	3C2 * 15C(10-2) / 43758 = 0.4412
3	3C3 * 15C(10-3) / 43758 = 0.1471

(please UPVOTE)