replacement): the Square Root Law
Chapter 7. Expected values, standard errors, variances and covariances
(exercises)
A:
1) You can choose: – in game A you win 12 dollars with a chance of 50% or win nothing with a chance of 50%;
– in game B you win 18 dollars with a chance of 30% or win nothing with a chance of 70%.
Which is better?
2) You can choose: – in game A you win 12 dollars with a chance of 50% or win nothing with a chance of 50%;
– in game B you win 20 dollars with a chance of 30% or win nothing with a chance of 70%.
Which is better?
3) You can choose: – in game A you may win 100 dollars with a chance of 30% – but win only 10 dollars with a chance of 70%;– in game B you may win 250 dollars with a chance of 10% – but win only 12 dollars with a chance of 90%.
Which is better?
3') You can choose: – in game A you may win 100 dollars with a chance of 30% – but win only 10 dollars with a chance of 70%;– in game B you may win 250 dollars with a chance of 10% – but win only 15 dollars with a chance of 90%.
Which is better?
4) You are at a decision point; decision A is thought to result in a gain of 100,000 dollars (with 10%
probability); or it may end in a gain of only 5,000 dollars (with 70% probability) – and there is a 20% chance that it will end in a 40,000 dollars loss.
With decision B there is a 25% chance for a gain of 20,000 dollars; 50% chance for a gain of 1,200 dollars; and 25% chance that your gain will be zero (but no loss either).
Which is better?
5) In a game your chance for winning nothing equals 90% – and you have a 10% chance for winning 10 dollars.
a) what is your expected gain in this game?
b) what is the S.D. of the gains in this game?
(Let the experiment be one such game, X denoting your net gain; (b) then asks for the S.E. of X.)
6) In a game your chance for winning nothing equals 99% – and you have a 1% chance for winning 100 dollars.
a) what is your expected gain in this game?
b) what is the S.D. of the gains in this game?
7) In a game your chance for winning nothing equals 50% – but you have a 50% chance for winning 2 dollars.
a) what is your expected gain in this game?
b) what is the S.D. of the gains in this game?
In what sense is it the best?
9) Which of the games in exercises 5–7 would you recommend – to somebody who is anxious, avoiding risks?
– to somebody who likes risks?
10) The values of a variable X are all positive. Can the S.D. (that shows the medium deviances from the average) be bigger than the average?
11) Ten million cards representing the heights of a ten millions population in a box (cards with numbers, one for each inhabitant of Midgetland with his height in centimeters on it). The population average in Midgetland is 120 cm, the S.D. of the heights in the population is 10 cm.
Experiment: 100 draws from the box, with replacement. Denote X the sum of the draws.
a) what is the average of the box? what is the S.D. of the box?
b) E(X)=?
c) var(X)=?
d) S.E.(X)=?
11') Ten million cards representing the heights of a ten millions population in a box (cards with numbers, one for each inhabitant of Midgetland with his height in centimeters on it). The population average in Midgetland is 120 cm, the S.D. of the heights in the population is 10 cm.
Experiment: 100 draws from the box, with replacement. Denote Y the mean of the draws.
a) what is the average of the box? what is the S.D. of the box?
b) E(Y)=?
c) var(Y)=?
d) S.E.(Y)=?
12) Box with numbers of average m and standard deviation d. Experiment: n draws from the box, with replacement. Denote X the sum of the draws.
a) find the expected value of the random variable X;
b) find the variance of the random variable X;
c) find the standard error of the random variable X.
Denote Y the mean of the draws.
a) find the expected value of the random variable Y;
b) find the variance of the random variable Y;
c) find the standard error of the random variable Y.
B:
1) Seven 1s, two 2s and one 9 in a box. Two draws are made, with replacement. Denote X1 the first draw, denote X2 the second draw.
a) find the expected value of X1.
c) find the expected value of the sum of the draws.
d) find the expected value of the product of the draws.
e) D2(X1)=?; D2(X2)=?
f) cov(X1, X2)=?
g) D2(X1+X2)=?
h) D(X1)=?; D(X2)=?
i) D(X1+X2)=?
2) Seven 1s, two 2s and one 9 in a box. Two draws are made, without replacement. Denote X1 the first draw, denote X2 the second draw.
a) find the expected value of X1. b) find the expected value of X2.
c) find the expected value of the sum of the draws.
d) find the expected value of the product of the draws.
e) D2(X1)=?; D2(X2)=?
f) cov(X1, X2)=?
g) D2(X1+X2)=?
h) D(X1)=?; D(X2)=?
i) D(X1+X2)=?
* * * *
3) The number 1 is glued onto two sides of an otherwise fair dice, the number 2 is glued onto two sides and the number 3 is glued onto two sides so that the identical numbers be opposite each other. (When a 1 shows, a 1 is below, etc.) Two rolls with this dice. Denote X1 the first roll, denote X2 the second roll.
a) find the expected value of X1. b) find the expected value of X2.
c) find the expected value of the sum of the draws.
d) find the expected value of the product of the draws.
e) find the covariance of X1 and X2 . f) find the variances of X1 and of X2 .
g) find the variance of the sum of the two rolls.
4) The number 1 is glued onto two sides of an otherwise fair dice, the number 2 is glued onto two sides and the number 3 is glued onto two sides such that the identical numbers be opposite each other. (When a 1 shows, a 1 is below, etc.) One roll with this dice. Denote X1 the number shown, denote X2 the number below.
a) find the expected value of X1. b) find the expected value of X2.
d) find the expected value of the product X1 * X2. e) find the covariance of X1 and X2.
f) find the variances of X1 and of X2 .
g) find the variance of the sum of the two rolls.
5) The number 1 is glued onto two sides of an otherwise fair dice, the number 2 is glued onto two sides and the number 3 is glued onto two sides so that '1's are opposite '3's and '2's are opposite '2's. (When a 1 shows, a 3 is below, etc.) Two rolls with this dice. Denote X1 the first roll, denote X2 the second roll.
a) find the expected value of X1. b) find the expected value of X2.
c) find the expected value of the sum of the draws.
d) find the expected value of the product of the draws.
e) find the covariance of X1 and X2 . f) find the variances of X1 and of X2 .
g) find the variance of the sum of the two rolls.
6) The number 1 is glued onto two sides of an otherwise fair dice, the number 2 is glued onto two sides and the number 3 is glued onto two sides so that '1's be opposite '3's and '2's are opposite '2's. (When a 1 shows, a 3 is below, etc.) One roll with this dice. Denote X1 the number shown, denote X2 the number below.
a) find the expected value of X1. b) find the expected value of X2.
c) find the expected value of the sum X1 + X2. d) find the expected value of the product X1 * X2. e) find the covariance of X1 and X2.
f) find the variances of X1 and of X2 .
g) find the variance of the sum of the two rolls.
* * * *
7) Two draws, without replacement, from a box with 10 balls (3 of gold and 7 of iron). Denote Y2 the number of golden balls among the draws. E(Y2)=?
7') Three draws, without replacement, from a box with 10 balls (3 of gold and 7 of iron). Denote Y3 the number of golden balls among the draws. E(Y3)=?
8) Two draws, with replacement, from a box with 10 balls (3 of gold and 7 of iron). Denote Y2 the number of golden balls among the draws. E(Y2)=?
8') Three draws, with replacement, from a box with 10 balls (3 of gold and 7 of iron). Denote Y3 the number of golden balls among the draws. E(Y3)=?
C:
1) The variable X is uniformly distributed over the interval [0; π / 2 ] .
b) find the expected value of the random variable X;
c) find the expected value of the random variable sin X d) find the expected value of the random variable X2 e) find the expected value of the random variable sin2 X
2) The values of the random variable X are from the interval [ 0; π / 2 ] . The density function here.
(Elsewhere it equals zero.)
/a) sketch the graph of the density function/
b) find the expected value of the random variable X;
c) find the expected value of the random variable sin X d) find the expected value of the random variable X2 e) find the expected value of the random variable sin2 X
3) The values of the random variable X are from the interval [0; π / 2 ] . Its density function is on this interval. Elsewhere it equals zero.
/a) sketch the graph of the density function/
b) find the expected value of the random variable X;
c) find the expected value of the random variable sin X d) find the expected value of the random variable X2 e) find the expected value of the random variable sin2 X
Compare the results of exercises 1-3 (points (b) among themselves, points (c) among themselves etc.). Explain the differences.
Readings
[bib_12] Statistics. Copyright © 1998. W.W.Norton & Co., New York, London. Chapter 17. D. Freedman, R.
Pisiani, and R. Purves.
[bib_13] Probability and Statistical Inference. R Bartoszynski and M Niewiadomska-Bugaj. Copyright © 1996.
John Wiley & Sons, New York, Chichester, Brisbane, Toronto, Singapore. Chapter 8..