# ssrs 2008 r2 barcode font Random Variables and Probability Distributions in Software Printing QR Code JIS X 0510 in Software Random Variables and Probability Distributions

CHAPTER 2 Random Variables and Probability Distributions
QR Code Reader In None
Using Barcode Control SDK for Software Control to generate, create, read, scan barcode image in Software applications.
Make Quick Response Code In None
Using Barcode generator for Software Control to generate, create Quick Response Code image in Software applications.
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
QR Code Decoder In None
Using Barcode recognizer for Software Control to read, scan read, scan image in Software applications.
Print QR Code In C#.NET
Using Barcode encoder for VS .NET Control to generate, create QR Code image in Visual Studio .NET applications.
where u is the angle that line OP (from O to any point P on the target) makes with the axis.
QR Generator In .NET Framework
Using Barcode creation for ASP.NET Control to generate, create QR Code image in ASP.NET applications.
QR Code ISO/IEC18004 Encoder In .NET
Using Barcode maker for VS .NET Control to generate, create QR Code image in Visual Studio .NET applications.
Fig. 2-26
QR Code JIS X 0510 Creator In Visual Basic .NET
Using Barcode generation for Visual Studio .NET Control to generate, create Quick Response Code image in Visual Studio .NET applications.
Printing GS1 - 12 In None
Using Barcode generator for Software Control to generate, create UPC-A Supplement 5 image in Software applications.
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
Encode Data Matrix In None
Using Barcode creator for Software Control to generate, create Data Matrix ECC200 image in Software applications.
EAN / UCC - 14 Generation In None
Using Barcode creation for Software Control to generate, create EAN / UCC - 14 image in Software applications.
0 and u
Generate Bar Code In None
Using Barcode printer for Software Control to generate, create barcode image in Software applications.
Make EAN13 In None
Using Barcode encoder for Software Control to generate, create EAN 13 image in Software applications.
p>4.
EAN / UCC - 14 Encoder In None
Using Barcode printer for Software Control to generate, create EAN / UCC - 14 image in Software applications.
Printing Code 3 Of 9 In None
Using Barcode generator for Font Control to generate, create USS Code 39 image in Font applications.
1, 0
Scan Barcode In None
Using Barcode reader for Software Control to read, scan read, scan image in Software applications.
Code128 Reader In Java
Using Barcode scanner for Java Control to read, scan read, scan image in Java applications.
Show that although any two of these random variables are independent, i.e., their marginal density function factors, all three are not independent.
Encoding Bar Code In Java
Using Barcode creator for Eclipse BIRT Control to generate, create barcode image in BIRT reports applications.
Code 3/9 Maker In None
Using Barcode encoder for Word Control to generate, create Code 3/9 image in Office Word applications.
ANSWERS TO SUPPLEMENTARY PROBLEMS
Code 128A Generation In Java
Using Barcode encoder for BIRT reports Control to generate, create Code 128 Code Set A image in BIRT applications.
EAN / UCC - 13 Generation In Visual Studio .NET
Using Barcode generator for Reporting Service Control to generate, create UCC - 12 image in Reporting Service applications.
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