Suppose we play the following game. Roll a 6 sided die, you are rewarded a dolla
ID: 3149730 • Letter: S
Question
Suppose we play the following game. Roll a 6 sided die, you are rewarded a dollar amount of whatever number turns up. Suppose the die is fair (each side is equally likely). Define the random variable X to be the amount you are rewarded after one roll.
1) What are the possible values the random variable, X, can take on? – 1.5 pts
2) Write out explicitly the probability mass function (pmf) for the random variable X. – 1.5 pts
3) Write out explicitly the cumulative distribution function (cdf) for the random variable X. – 2 pts
4) Find E[X] – 0.5 pts
5) Find Var[X] – 1 pt
Explanation / Answer
when a rice is rolled
sample space = {1,2,3,4,5,6}
a) the possible value which x can take
dice number x
1 1
2 2
3 3
4 4
5 5
6 6
2) probability mass function
as the dice is fair therefore probability of each number = 1/6
therefore
x p(x)
1 1/6
2 1/6
3 1/6
4 1/6
5 1/6
6 1/6
3) the cumulative function will be
x = 1 p(x) = 1/6
x<=2 p(x) = 2/6
x<=3 P(x) = 3/6
x<=4 p(x) = 4/6
x<=5 p(x) = 5/6
x<=6 p(x) = 6/6
4) E(X) = X1*P(X1)+X2*P(X2)+.....+XN*P(XN)
= 1/6*1+1/6*2+3*1/6+4*1/6+5*1/6+6*1/6
= 21/6 = 3.5
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