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3.62. (a) 7 > 2 (b) 15 > 4 (c) !15>2
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35 (b) m1 2, mr 3
0, m2 6, mr 4
3 > 80, m3 24
1 (b) m 2, m4 1)(b
1, mr2 33 a) (b) [1
3.68. (a) (bk 3.70. peiva
ak 1) > (k qeivb
( 1)k](b 3.72. (e2iv
a)k > 2k 2ive2iv
1) 1)>2v2
3.71. ( sin av)>av
CHAPTER 3 Mathematical Expectation
3.75. (a) 11 > 144 (b) 11 > 144 (c) !11>12 (d) !11>12 (e) 1 > 144 (f) 1 > 11 3.76. (a) 1 (b) 1 (c) 1 (d) 1 (e) 0 (f) 0 3.77. (a) 73 > 960 (b) 73 > 960 (c) !73>960 (d) !73>960 (e) 1 > 64 (f) 15 > 73 3.78. (a) 233 > 324 (b) 233 > 324 (c) !233>18 (d) !233>18 (e) 91 > 324 (f) 91 > 233 3.79. (a) 4 (b) 4> !35 3.81. (a) (3x 2) > (6x 3.80. 3) for 0 !15>4 x 0 (b) 1 (b) (3y 2) > (6y 3) for 0 y 1
3.82. (a) 1 > 2 for x 3.83. (a)
0 (b) 1 for y
X E(Y u X)
0 4>3
2 5>7
Y E(X u Y)
0 4>3
1 7>6
2 1>2
3.84. (a)
6x2 6x 1 18(2x 1)2
for 0
1 (b)
6y2 6y 1 18(2y 1)2
for 0
3.85. (a) 1 > 9 (b) 1 3.86. (a) (b)
X Var(Y u X)
0 5>9
1 4>5
2 24 > 49
Y Var(X u Y)
0 5>9
1 29 > 36
2 7 > 12
3.87. (a) 1 > 2 (b) 2 (useless) 3.92. (a) 0 (b) ln 2 (c) 1
3.89. (a) e 2 (b) 0.5 3.93. (a) 1> !3 (b) #1 (1> !2) (c) 8 > 15
3.94. (a) does not exist (b) 1 (c) 0 3.96. (a) 1 3.97. (a) #1
1 2 !3
3.95. (a) 3 (b) 3 (c) 3
(b) 1 > 2 (23>2) (c) !1>2 (d) #1 (1> !10)
(3> !10) (b) #1
3.98. (a) 1 (b) (!3
1)>4 (c) 16 > 81 3.100. (a) 1 2e 1 !3)>3 3.104. (a) 2 (b) 9 2e 1 (b) does not exist
3.99. (a) 1 (b) 0.17 (c) 0.051 3.101. (a) (5 2!3)>3 (b) (3
3.102. (a) 2 (b) 9
3.103. (a) 0 (b) 24 > 5a
CHAPTER 3 Mathematical Expectation
3.105. (a) 7 > 3 (b) 5 > 9 (c) (et 3.106. (a) 1 > 3 (b) 1 > 18 (c) 2(et 3.107. (a) 21 > 2 (b) 35 > 4 3.108. (a) 4 > 3 3e3t) > 6 (d) (eiv t) > t2 (d) 2(eiv 2e2iv 1 3e3iv)>6 7 > 27
2e2t 1
iv)>v2 (e) 1 > 135
(b) 2 > 9 (c) (1 2te2t 2!18>15 (f) 12 > 5
1)>15
e2t) > 2t2
2ive2iv
e2iv)>2v2
3.109. (a) 1 (b) 8(2 !2 3.110. (a) 2 (b) !2p>2 3.111. (a) 0 (b) 1 > 3 (c) 0
Special Probability Distributions
The Binomial Distribution
Suppose that we have an experiment such as tossing a coin or die repeatedly or choosing a marble from an urn repeatedly. Each toss or selection is called a trial. In any single trial there will be a probability associated with a particular event such as head on the coin, 4 on the die, or selection of a red marble. In some cases this probability will not change from one trial to the next (as in tossing a coin or die). Such trials are then said to be independent and are often called Bernoulli trials after James Bernoulli who investigated them at the end of the seventeenth century. Let p be the probability that an event will happen in any single Bernoulli trial (called the probability of success). Then q 1 p is the probability that the event will fail to happen in any single trial (called the probability of failure). The probability that the event will happen exactly x times in n trials (i.e., successes and n x failures will occur) is given by the probability function
f (x) P(X x) n a b pxqn x
n! x!(n x)!
pxqn
where the random variable X denotes the number of successes in n trials and x
EXAMPLE 4.1
0, 1, . . . , n.
The probability of getting exactly 2 heads in 6 tosses of a fair coin is 6 1 1 a ba b a b 2 2 2
2 6 2
6! 1 1 a b a b 2!4! 2 2
15 64
The discrete probability function (1) is often called the binomial distribution since for x corresponds to successive terms in the binomial expansion
0, 1, 2, . . . , n, it
p) n
n a bqn 1p 1
n a bqn 2p2 2
n a a x b p x qn x 0
The special case of a binomial distribution with n
1 is also called the Bernoulli distribution.
Some Properties of the Binomial Distribution
Some of the important properties of the binomial distribution are listed in Table 4-1.
CHAPTER 4 Special Probability Distributions
Table 4-1 Mean Variance Standard deviation Coefficient of skewness Coefficient of kurtosis Moment generating function Characteristic function s a3
np npq !npq q p !npq
a4 M(t) f(v)
3 (q (q
6pq npq pet)n peiv)n
(100) A 2 B
EXAMPLE 4.2 In 100 tosses of a fair coin, the expected or mean number of heads is m 2(100) A 1 B A 1 B 5. standard deviation is s 2 2
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