Post on 01-Apr-2015
Christopher Dougherty
EC220 - Introduction to econometrics (review chapter)Slideshow: exercise r.2
Original citation:
Dougherty, C. (2012) EC220 - Introduction to econometrics (review chapter). [Teaching Resource]
© 2012 The Author
This version available at: http://learningresources.lse.ac.uk/141/
Available in LSE Learning Resources Online: May 2012
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1
EXERCISE R.2
R.2* A random variable X is defined to be the larger of the numbers when two dice are thrown, or the number if they are the same. Find the probability distribution for X.
2
red 1 2 3 4 5 6 green
1
2
3
4
5
6
Suppose that one die is red and the other green.
EXERCISE R.2
3
EXERCISE R.2
red 1 2 3 4 5 6 green
1
2
3
4
5
6 6
Then, for example, if the red die is 4 and the green one is 6, X is equal to 6.
4
EXERCISE R.2
red 1 2 3 4 5 6 green
1
2
3
4
5 5
6
Similarly, if the red die is 2 and the green one is 5, X is equal to 5.
5
EXERCISE R.2
red 1 2 3 4 5 6 green
1 1 2 3 4 5 6
2 2 2 3 4 5 6
3 3 3 3 4 5 6
4 4 4 4 4 5 6
5 5 5 5 5 5 6
6 6 6 6 6 6 6
The table shows all the possible outcomes.
6
X 123456
EXERCISE R.2
red 1 2 3 4 5 6 green
1 1 2 3 4 5 6
2 2 2 3 4 5 6
3 3 3 3 4 5 6
4 4 4 4 4 5 6
5 5 5 5 5 5 6
6 6 6 6 6 6 6
If you look at the table, you can see that X can be any of the numbers from 1 to 6.
7
X f 123456
EXERCISE R.2
red 1 2 3 4 5 6 green
1 1 2 3 4 5 6
2 2 2 3 4 5 6
3 3 3 3 4 5 6
4 4 4 4 4 5 6
5 5 5 5 5 5 6
6 6 6 6 6 6 6
We will now define f, the frequencies associated with the possible values of X.
8
X f 1234 756
EXERCISE R.2
red 1 2 3 4 5 6 green
1 1 2 3 4 5 6
2 2 2 3 4 5 6
3 3 3 3 4 5 6
4 4 4 4 4 5 6
5 5 5 5 5 5 6
6 6 6 6 6 6 6
For example, there are seven outcomes which make X equal to 4.
9
X f 1 12 33 54 75 96 11
EXERCISE R.2
red 1 2 3 4 5 6 green
1 1 2 3 4 5 6
2 2 2 3 4 5 6
3 3 3 3 4 5 6
4 4 4 4 4 5 6
5 5 5 5 5 5 6
6 6 6 6 6 6 6
Similarly you can work out the frequencies for the other values of X.
10
X f p 1 12 33 54 75 96 11
EXERCISE R.2
red 1 2 3 4 5 6 green
1 1 2 3 4 5 6
2 2 2 3 4 5 6
3 3 3 3 4 5 6
4 4 4 4 4 5 6
5 5 5 5 5 5 6
6 6 6 6 6 6 6
Finally we will derive the probability of obtaining each value of X.
11
X f p 1 12 33 54 75 96 11
EXERCISE R.2
red 1 2 3 4 5 6 green
1 1 2 3 4 5 6
2 2 2 3 4 5 6
3 3 3 3 4 5 6
4 4 4 4 4 5 6
5 5 5 5 5 5 6
6 6 6 6 6 6 6
If there is 1/6 probability of obtaining each number on the red die, and the same on the green die, each outcome in the table will occur with 1/36 probability.
12
X f p 1 1 1/362 3 3/363 5 5/364 7 7/365 9 9/366 11 11/36
Hence to obtain the probabilities associated with the different values of X, we divide the frequencies by 36.
EXERCISE R.2
red 1 2 3 4 5 6 green
1 1 2 3 4 5 6
2 2 2 3 4 5 6
3 3 3 3 4 5 6
4 4 4 4 4 5 6
5 5 5 5 5 5 6
6 6 6 6 6 6 6
Copyright Christopher Dougherty 1999–2006. This slideshow may be freely copied for personal use.
26.08.06