wo fair cubes are rolled. The random variable X represents the difference between the values of the two cubes.
a) Find the mean of this probability distribution. (i.e. Find E[X] )
b) Find the variance and standard deviation of this probability distribution.
(i.e. Find V[X] and SD[X])
The random variables A and B are defined as follows:
A = X-10 and B = [(1/2)X]-5
c) Show that E[A] and E[B].
d) Find V[A] and V[B].
e) Arnold and Brian play a game using two fair cubes. The cubes are rolled, and Arnold records his score using the random variable A and Brian uses the random variable B. They repeat this for a large number of times and compare their scores. Comment on any likely differences or similarities of their scores.
probability distribution:
values on cubes difference probability
1,1 0 6/36=1/6
1,2 1 10/36=5/18
2,1 1 10/36=5/18
2,2 0 6/36=1/6
2,3 1 10/36=5/18
3,2 1 10/36=5/18
3,3 0 6/36=1/6
3,4 1 10/36=5/18
4,3 1 10/36=5/18
4,4 0 6/36=1/6
4,5 1 10/36=5/18
5,4 1 10/36=5/18
5,5 0 6/36=1/6
5,6 1 10/36=5/18
6,5 1 10/36=5/18
6,6 0 6/36=1/6
1,3 2 8/36=2/9
3,1 2 8/36=2/9
4,2 2 8/36=2/9
2,4 2 8/36=2/9
5,3 2 8/36=2/9
3,5 2 8/36=2/9
4,6 2 8/36=2/9
6,4 2 8/36=2/9
1,4 3 6/36=1/6
4,1 3 6/36=1/6
5,2 3 6/36=1/6
2,5 3 6/36=1/6
3,6 3 6/36=1/6
6,3 3 6/36=1/6
5,1 4 4/36=1/9
1,5 4 4/36=1/9
2,6 4 4/36=1/9
6,2 4 4/36=1/9
1,6 5 2/36=1/18
6,1 5 2/36=1/18
a)
"\\mu=E(X)=\\sum x_ip_i="
"=10\\cdot5\/18+2\\cdot8\\cdot2\/9+3\\cdot6\\cdot1\/6+4\\cdot4\\cdot1\/9+5\\cdot2\\cdot1\/18=11.67"
b)
"V(X)=\\sum p_i (x_i-\\mu)^2=11.67^2+10\\cdot5\\cdot10.67^2\/18+8\\cdot2\\cdot9.67^2\/9+"
"+8.67^2+4\\cdot7.67^2\/9+2\\cdot6.67^2\/18=724.93"
"SD(X)=\\sqrt{V(X)}=\\sqrt{724.93}=26.92"
c)
A = X-10 and B = [(1/2)X]-5
"E[A]=E[X]-10"
"E[B]=E[X]\/2-5"
"E[A]=2E[B]"
d)
"V[A]=E[A^2]-(E[A])^2=E[(X-10)^2]-(E[X])^2+20E[X]-100="
"=E[X^2]-20E[X]+100-(E[X])^2+20E[X]-100=E[X^2]-(E[X])^2=V[X]"
"V[B]=E[B^2]-(E[B])^2=E[A^2\/4]-(E[A])^2\/4=V[X]\/4"
e)
In probability theory, the law of large numbers is a theorem that describes the result of performing the same experiment a large number of times. According to the law, the average of the results obtained from a large number of trials should be close to the expected value and will tend to become closer to the expected value as more trials are performed.
So, for a large number of times scores of Arnold and Brian will be the same.
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