Consider rolling a 6-sided die. The outcome of interest is the number of dots th

Question

Consider rolling a 6-sided die. The outcome of interest is the number of dots that appears on the topside when the die stops rolling. The possible outcomes are 1, 2, 3, 4, 5 & 6. Suppose we play the following game. Roll the 6-sided die, you are rewarded a dollar amount equal to the number of dots on the topside of the die when it comes to rest. 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 Answer the following questions: 1) what are the possible values the random variable, X, can take on?-1.5 pts 2) Explicitly write out the probability mass function (pmf) for the random variable X.-1.75 pts 3) Write out explicitly the cumulative distribution function (cd for the random variable X. 2 pts 4) What is the typical' amount rewarded when playing this game? 0.75 pts 5) How much does the amount rewarded vary around the value obtained in part (4)? 1 pts

Explanation / Answer

1) possible values X can take =-{1,2,3,4,5,6}

2)as each of 6 number is equally likely to appear ; therefore

probability mass function : P(X=x) =1/6

3) cumulative distirbution fnction F(x) =P(X<=x) =x/6

4)

from above distribution

typical value E(X) =3.5

5) amount of variation around typical value =std deviaiton =1.708

x P(x) xP(x) x2P(x) 1    1/6 0.1667 0.1667 2    1/6 0.3333 0.6667 3    1/6 0.5000 1.5000 4    1/6 0.6667 2.6667 5    1/6 0.8333 4.1667 6    1/6 1.0000 6.0000 total 3.5 15.166667 E(X) = xP(x) = 3.5 E(X2) = x2P(x) = 15.17 Var(X)=2 E(X2)-(E(X))2= 2.917 std deviation = 2 = 1.708

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