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Expectation of random variables 3.1. In a lottery there are 200 prizes of \$5, 20 prizes of \$25, and 5 prizes of \$100. Assuming that 10,000 tickets are to be issued and sold, what is a fair price to pay for a ticket
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Let X be a random variable denoting the amount of money to be won on a ticket. The various values of X together with their probabilities are shown in Table 3-2. For example, the probability of getting one of the 20 tickets giving a \$25 prize is 20 > 10,000 0.002. The expectation of X in dollars is thus E(X) (5)(0.02) (25)(0.002) (100)(0.0005) (0)(0.9775) 0.2
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or 20 cents. Thus the fair price to pay for a ticket is 20 cents. However, since a lottery is usually designed to raise money, the price per ticket would be higher. Table 3-2 x (dollars) P(X x) 5 0.02 25 0.002 100 0.0005 0 0.9775
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3.2. Find the expectation of the sum of points in tossing a pair of fair dice.
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Let X and Y be the points showing on the two dice. We have E(X) Then, by Theorem 3-2, E(X E(Y) 1 1 6 Y) 1 2 6 c 1 6 6
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3.3. Find the expectation of a discrete random variable X whose probability function is given by f (x)
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We have E(X)
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To find this sum, let
1 ax 2
1 2
1, 2, 3, c )
1 2 4 1 3 8
Then Subtracting, Therefore, S 2.
1 S 2 1 S 2 1 2
1 2 4 1 4 1 4
c 1 16 1 16 1 16 c c c 1
1 3 8 1 8
1 2 8
CHAPTER 3 Mathematical Expectation
3.4. A continuous random variable X has probability density given by f (x) Find (a) E(X), (b) E(X2).
` ` ` 2x) dx
2e 0
E(X)
E(X2)
3.5. The joint density function of two random variables X and Y is given by f (x, y) e xy>96 0 0 x 4, 1 otherwise 3Y).
2 B(x2)
2 B (x)
3 `xf (x) dx e
2 3 `x f (x) dx 2x
30 x(2e (1)
2 3 x2e
2x dx
R 2
2 3 xe
0 ` 0
2x dx
(2x)
(2)
R 2
Find (a) E(X), (b) E(Y), (c) E(XY), (d) E(2X
(a) (b) (c)
E(X) E(Y) E(XY)
3 `xf (x, y) dx dy `
3 `yf (x, y) dx dy `
3 `(xy) f (x, y) dx dy `
5 xy 3y 1y 96 dx dy 0 4 5
E(2X
3 `(2x `
3y) f (x, y) dx dy
5 xy 3y 1(xy) 96 dx dy 0
5 xy 3y 1x 96 dx dy 0
8 3 31 9 248 27
3 (2x 0 y 1
Another method (c) Since X and Y are independent, we have, using parts (a) and (b), E(XY) E(X)E(Y) 8 31 3 9 8 2 3 248 27 3 31 9
3y)
xy dx dy 96
47 3
(d) By Theorems 3-1 and 3-2, pages 76 77, together with (a) and (b), E(2X 3Y) 2E(X) 3E(Y)
47 3
3.6. Prove Theorem 3-2, page 77.
Let f (x, y) be the joint probability function of X and Y, assumed discrete. Then E(X Y) a a (x
y) f (x, y) a a yf (x, y)
a a xf (x, y)
E(X)
E(Y)
If either variable is continuous, the proof goes through as before, with the appropriate summations replaced by integrations. Note that the theorem is true whether or not X and Y are independent.
CHAPTER 3 Mathematical Expectation
3.7. Prove Theorem 3-3, page 77.
Let f (x, y) be the joint probability function of X and Y, assumed discrete. If the variables X and Y are independent, we have f (x, y) f1 (x) f2 ( y). Therefore, E(XY) a a xyf (x, y)
a [(xf1(x)E( y)]
E(X)E(Y)
a B xf1(x) a yf2( y) R
x x y
a a xyf1(x) f2 ( y)
If either variable is continuous, the proof goes through as before, with the appropriate summations replaced by integrations. Note that the validity of this theorem hinges on whether f (x, y) can be expressed as a function of x multiplied by a function of y, for all x and y, i.e., on whether X and Y are independent. For dependent variables it is not true in general.
Variance and standard deviation 3.8. Find (a) the variance, (b) the standard deviation of the sum obtained in tossing a pair of fair dice.
(a) Referring to Problem 3.2, we have E(X) E(X2) Then, by Theorem 3-4, Var (X) E(Y2) 1 12 6 E(Y) 1 22 6 91 6 1 > 2. Moreover, c 1 62 6 35 12
91 6
Var (Y)
and, since X and Y are independent, Theorem 3-7 gives Var (X Y) Var (X) 2Var (X
7 2
Var (Y) 35 A6
35 6