Question

A student makes a trip once per day to the store, and he always buys a snack. The student eats the snack 70% of the time, but the other 30% of the time, his roommate eats it first.

a. During a period of four days, the student keeps track of whether he gets to eat his snack. What is the sample space of possible outcomes?

b. Let X be the number of times, within the four day period, that he gets to eat his snack. What is the mass of X?

c. What is the CDF of X?

Answer 1

(a)

Sample Space of possible outcomes:

S = {0,1,2,3,4}

(b)

Probability Mass Function of X is given by:

For X = 0; Probability that he does not get in all 4 days = 0.30⁴ = 0.0081

For X = 1. Probability that he does not get in 1 day & gets in other 3 days out of 4 days = 4 X 0.70 X 0.30³ = 0.0756

For X = 2. Probability that he does not get in 2 days & gets in other 2 days out of 4 days = 6 X 0.70² X 0.30² = 0.2646

For X = 3. Probability that he does not get in 3 days & gets in other 1 day out of 4 days = 4 X 0.70³ X 0.30 = 0.4116

For X = 4. Probability that he gets in all 4 days = 0.70⁴ = 0.2401

So,

the probability Mass Function of X is given by:

x	f(x)
0	0.0081
1	0.0756
2	0.2646
3	0.4116
4	0.2401

(c)

CDFof X is given by:
F(x) = 0 for x < 0

      = 0.0081 for 0 $\leq$ x < 1

        = 0.0081 + 0.0756 = 0.0837 for 1 $\leq$ x < 2

        = 0.0837 + 0.2646 = 0.3483 for 2 $\leq$ x < 3

       = 0.3483 + 0.4116 = 0.7599 for 3 $\leq$ x < 4

      = 1 for x $\geq$ 4