概率论与数理统计英文版3

Then the distribution function of X is (Figure 3.1.1)
⎧0 ,
F
(x)
=
⎪⎪ ⎨ ⎪
n 6
,
⎪⎩1,
if x < 1; if n ≤ x < n +1, n = 1, 2, ,5; if x ≥ 6.
Figure 3.1.1 The distribution function in Example 3.1.3
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Example 1
For an experiment in which a coin is tossed three times (or 3 coins are
tossed once), construct the distribution of X. (Let X denote the number of
1. random variable definition
Definition 3.1.1 A random variable is a real valued function defined on a sample space; i.e. it assigns a real number to each sample point in the sample space. Here are some examples. Example 3.1.1 A fair die is tossed. The number X shown is a random variable, it takes values in the set {1, 2, 6}. Example 3.1.2 The life t of a bulb selected at random from bulbs produced by company A is a random variable, it takes values in the interval (0 ,∞) .
Since the outcomes of a random experiment can not be predicted in advance, the exact value of a random variable can not be predicted before the experiment, we can only discuss the probability that it takes some
击，如果不命中则一直射到子弹用尽，求耗用子弹数 x 的概率分布。 *****
Example 3.2.1 A die is tossed, by X we denote the number shown, Assume that the probability P(X = k) is proportional to k , k = 1, 2, ,6 . Find the probability distribution of X . Solution Assume that
F(X ) = P(X ≤ x) .
x∈ R
is called the distribution function of X
Note The distribution function F(X ) is defined on real numbers, not on
sample space.
Example 3.1.3 Let X be the number we get from tossing a fair die.
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value or the values in some subset of R.
2. Distribution function
Definition 3.1.2 Let X be a random variable on the sample space S .
Then the function
3. Random Variables
3.1 Definition of Random Variables In engineering or scientific problems, we are not only interested in
the probability of events, but also interested in some variables depending on sample points. （定义在样本点上的变量）
company A , assume the distribution function of X is (in hours)
F ( x)
=
P( X
≤
x)
=
⎧⎪⎨1
−
−
e
x 2000
,
⎪⎩ 0,
x ≥ 0; x < 0.
Find P(X ≤ 2000) , P(1000 < X ≤ 3000) .
Solution
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In this book, we study two kinds of random variables.
Definition 3.2.1 A random variable X is called a discrete random
variable, if it takes values from a finite set or, a set whose elements can
(1) Pi ≥0
(2) P1+P2+…+Pn=1
离散型分布函数
And the distribution function of X is given by
F (x) = P( X ≤ x) = ∑ pn an ≤ x
(3.2.2)
In general, it is more convenient to use (3.2.1) instead of (3.2.2). Equation (3.2.1) is called the probability distribution of the discrete random variable X .
P( X1
=
k)
=
k −1 36
,
And
k = 2,3, 4,5
∪5
{X = −1} = {X1 = k}
k =2
so
Thus
∑ P( X = −1) = 5 P( X = k) = 5
k =2
18
P( X = 1) = 1− P( X = −1) = 13 18
⎧0,
F ( x)
=
P( X
≤
x)
=
F
( x0
+
0)
=
F
( x0
)
.This
is
to
say, the
distribution function F(x) of a random variable X is right continuous.
Example 3.1.4 Let X be the life of automotive parts produced by
Question： What are the probabilities P(X < 2000) and P(X = 2000) ?
Example 3.1.5 A player tosses two fair dice, if the total number shown
is 6 or more, the player wins $1, otherwise loses $1. Let X be the
be written as a sequence {a1, a2, an, }
Assume a discrete random variable X takes values from the set
X = {a1, a2, an , } . Let
P( X = an ) = pn , n = 1, 2, .
(3.2.1)
Then we have pn ≥ 0 , n = 1, 2, , ∑ pn = 1. n
the probability distribution of the discrete random variable X (概率分
布)
X
a1
a2
an
probability p1
p2
pn
注意随机变量 X 的分布所满足的条件
P(X = k) = ck , c = constant, k = 1, 2, , 6 . Since the events P(X = k) , k = 1, 2, ,6 are mutually exclusive and their union is the certain event, i.e., the sample space S , we have
For example, we maybe interested in the life of bulbs produced by a certain company, or the weight of cows in a certain farm, etc. These ideas lead to the definition of random variables.
head occurrence)
Solution
n=3, p=1/2
X
pr
0
1/8
1
3/8
2
3/8
3
1/8
Example 2 在一次试验中，事件 A 发生的概率为 p, 不发生的概率为 1－p, 用 X=0 表示事件 A 没有发生，X=1 表示事件 A 发生，求 X 的分布。 two-point distribution(两点分布)