probability math

1. A random variable X results from an experiment in which a fair 1 6-sided die is rolled at the same time a fair coin is flipped. Specifically, X = {the number of dots showing on the die + Number of heads showing on the coin}.

a) Determine all possible values of X and the probability of occurrence for each of these values (i.e., the pmf of X). Show your work or justify your answer.

b) Plot the CDF F(x) of X. Clearly label all values (on both axes).

c) Determine the mean of X. Show your work.

2. Let X be a continuous random variable with the following probability density function (pdf):

given mathematically by

f ( x )={ 2 3

x , 0≤ x≤1

1− x 3

, 1< x≤3

0, Otherwise } .

a) Determine P{X < 1}.

b) Calculate the mean of X.

b) Calculate the variance of X.

0

0.1

0.2

0.3

0.4

0.5

0.6

0 0.5 1 1.5 2 2.5 3