ssrs 2008 r2 barcode font Random Variables and Probability Distributions in Software

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CHAPTER 2 Random Variables and Probability Distributions
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where r is the distance from the axis OB. Show that the distribution of particles along the target is given by g(u) e cos u 0 0 u p>2 otherwise
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where u is the angle that line OP (from O to any point P on the target) makes with the axis.
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2.102. In Problem 2.101 find the probability that a particle will hit the target between u 2.103. Suppose that random variables X, Y, and Z have joint density function f (x, y, z) e 1 0 cos px cos py cos pz 0 x 1, 0 otherwise y
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p>4.
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Show that although any two of these random variables are independent, i.e., their marginal density function factors, all three are not independent.
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2.38. x f (x) 2.40. 0 1>8 1 3>8 2 3>8 3 1>8
x f (x)
0 3 > 28
1 15 > 28
2 5 > 14
x f (x)
0 9 > 64
1 15 > 32
2 25 > 64
x f (x)
0 194,580 270,725
1 69,184 270,725
2 6768 270,725
3 192 270,725
4 1 270,725
x f (x)
0 1>8
1 1>2
2 7>8
2.46. (a)
x f (x)
1 1>8
2 1>4
3 3>8
4 1>4
(b) 3 > 4 (c) 7 > 8 (d) 3 > 8 (e) 7 > 8
CHAPTER 2 Random Variables and Probability Distributions
e 1 0 e
2.47. (a) 3 (b) e
(c) e
(d) 1
2.48. F (x) 0 (2x 3 (3x 2 1
x x x x x x
0 0 1 2 3 3
2.49. (a) 6 > 29 (b) 15 > 29 (c) 19 > 116
2.50. F (x)
2)>29 2)> 29
2.51. (a) 1/27 (b) f (x)
x 2/9 0
0 x 3 (c) 26 > 27 (d) 7 > 27 otherwise 0 x 2 >4 1 x x x 0 2 2
2.53. (a) 1 > 2
(b) 1 > 2 (c) 3 > 4 (d) F(x)
2.54. (a) 1 > 36 (b) 1 > 6 (c) 1 > 4 (d) 5 > 6 (e) 1 > 6 (f) 1 > 6 (g) 1 > 2 2.55. (a) f1(x) e x>6 0 x 1, 2, 3 (b) f2( y) other x
y>6 0
y 1, 2, 3 other y
2.56. (a) 3 > 2 (b) 1 > 4 (c) 29 > 64 (d) 5 > 16 0 1 (x 3 2 1 x x x 0 1 1
2.57. (a) F1(x)
(b) F2( y)
0 1 (y 3 2 1
y y y
0 1 1
2.58. (a) f (x u y) (b) f ( y u x) 2.59. (a) f (x u y) (b) f ( y u x)
f1(x) for y f2( y) for x e (x 0 (x 0 (x 2 0 (x 2 0 e 0
1, 2, 3 (see Problem 2.55) 1, 2, 3 (see Problem 2.55)
1 2)
y)>( y
0 x 1, 0 other x, 0 y 0 0
1 3)
y)>(x
1 2)
1, 0 y 1, other y
2.60. (a) f (x u y) (b) f ( y ux)
y 2)>( y 2
0 x 1, 0 other x, 0 y 0 0 x x
y 2)>(x 2
1 3)
1, 0 y 1, other y e e 0
2.61. (a) f (x u y) 2.62. e
1y >2 !y
0, y 0, y
0 (b) f (y u x) 0 2.64. (2p)
0, y 0, y for y
0 0 0; 0 otherwise
for y y
0; 0 otherwise
1> 2y 1> 2 e y> 2
2.66. 1>p for 2.68. (a) g( y)
p>2 e
p>2; 0 otherwise
1 6 (1 1(y 6
5 y 1 (b) g( y) otherwise 0, v
y) 1)
2>3 2>3
0 y 1 1 y 2 otherwise
2.70. ve v >(1
u)2 for u
0; 0 otherwise
CHAPTER 2 Random Variables and Probability Distributions
e ln z 0 u
2.73. g(z)
0 z 1 otherwise 0 u 1 1 u 2 otherwise u u e e 0 1 1 0 0 0
2.77. g(x)
x 3e x/6 0
2.74. g(u)
u 2 0 e ue 0
2.78. 1 > 4
2.75. g(u)
2.79. 61 > 72 x y 1 x 1) (d) 26 > 81 (e) 1 > 9
2.81. (a) 2 (b) F(x)
y x x
1; y
1, 2, 3, c
2.82. (a) 4 (b) F(x) 2.83. (a) 3 > 7 2.86. (a) c1 2.88. (a) 1 > 4 2.90. (a) e e 0
2x (2x
0 (d) 3e 0
(e) 5e 2e
(b) 5 > 7
2, c2
2.84. (a) c
1 (b) e
9 (b) 9e
(c) 4e
(d) e
(e) 4e
1 (y e4 0
(b) 27 > 64 (c) f1(x)
2y/ !y
0 x 1 (d) f2(y) otherwise
u 0 otherwise
0 y 2 otherwise
18e y 0 (b) e otherwise 0 1 6 0 0 1 ln 2 2
1 2.91. (b) (1 2 2.93. g(z) e2 0
ln 2) (c) e
z> 2
(d) ln 2
2.95. (b) 15 > 256 (c) 9 > 16
(d) 0
2.100. (a) 45 > 512
(b) 1 > 14
2.94. (b) 7 > 18
2.102. !2>2
CHAPTER 12 CHAPTER 3
Mathematical Expectation
Definition of Mathematical Expectation
A very important concept in probability and statistics is that of the mathematical expectation, expected value, or briefly the expectation, of a random variable. For a discrete random variable X having the possible values x1, c , xn, the expectation of X is defined as E(X) or equivalently, if P(X xj) E(X) x1P(X f (xj), x1 f (x1 ) c
xn P(X
xn )
a xj P(X
xn f(xn)
a xj f(xj)
a x f(x)
where the last summation is taken over all appropriate values of x. As a special case of (2), where the probabilities are all equal, we have E(X) x1 x2 n c xn (3)
which is called the arithmetic mean, or simply the mean, of x1, x2, c , xn. If X takes on an infinite number of values x1, x2, c , then E(X) g ` 1 xj f(xj) provided that the infinite sej ries converges absolutely. For a continuous random variable X having density function f(x), the expectation of X is defined as
E(X)
3 `x f (x) dx
provided that the integral converges absolutely. The expectation of X is very often called the mean of X and is denoted by mX, or simply m, when the particular random variable is understood. The mean, or expectation, of X gives a single value that acts as a representative or average of the values of X, and for this reason it is often called a measure of central tendency. Other measures are considered on page 83.
EXAMPLE 3.1 Suppose that a game is to be played with a single die assumed fair. In this game a player wins $20 if a 2 turns up, $40 if a 4 turns up; loses $30 if a 6 turns up; while the player neither wins nor loses if any other face turns up. Find the expected sum of money to be won. Let X be the random variable giving the amount of money won on any toss. The possible amounts won when the die turns up 1, 2, c, 6 are x1, x2, c, x6, respectively, while the probabilities of these are f(x1), f (x2), . . . , f (x6). The probability function for X is displayed in Table 3-1. Therefore, the expected value or expectation is
E(X)
1 (0) 6
1 (20) 6
1 (0) 6
1 (40) 6
1 (0) 6
1 ( 30) 6
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