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6. 设

X

有连续的尾分布函数

G(x)=P(X>x)

, 定义它的广义逆

G^{-1}(u):=\inf \{x: G(x) \leq

$u\} ...

Apr 8, 2024

6. 设

X

有连续的尾分布函数 G(x)=P(X>x), 定义它的广义逆

G^{-1}(u):=\inf \{x: G(x) \leq

u\} ; U \sim U(0,1) ; g

是

[0,1]

上的严格单调增函数,

g(0)=0, g(1)=1

, 它的逆记为

g^{-1}

; (1) 证明

G^{-1}(U) \stackrel{d}{=} X

; (2) 求

G(X)

的分布; (3) 设

Y:=g(G(X))

, 求

Y

的尾分布函数; (4) 试用

X, G^{-1}, g^{-1}

构造随机变量

Y

, 使得

Y

的尾分布函数为

g \circ G

Solution by Steps

step 1

To prove

G^{-1}(U) \stackrel{d}{=} X

, we need to show that for any

t

P(G^{-1}(U) \leq t) = P(X \leq t)

step 2

Since

U \sim U(0,1)

, for any

t

P(G^{-1}(U) \leq t) = P(U \leq G(t))

step 3

By definition of

G

, G(t) = P(X > t) , so P(U \leq G(t)) = 1 - P(X > t) = P(X \leq t)

step 4

Therefore,

G^{-1}(U) \stackrel{d}{=} X

is proven

Answer

G^{-1}(U) \stackrel{d}{=} X

is proven.

Key Concept

Quantile Function

Explanation

The quantile function

G^{-1}

applied to a uniform random variable

U

(0,1)

yields a random variable with the same distribution as

X

step 1

To find the distribution of

G(X)

, we calculate

P(G(X) \leq u)

for

u \in [0,1]

step 2

By definition of

G

G(X) \leq u

implies P(X > x) \leq u where

x = G^{-1}(u)

step 3

Thus, P(G(X) \leq u) = P(X > G^{-1}(u)) = 1 - G(G^{-1}(u)) = 1 - u

step 4

The distribution of

G(X)

is uniform on

[0,1]

Answer

The distribution of

G(X)

is uniform on

[0,1]

Key Concept

Distribution of a Transformed Random Variable

Explanation

The distribution of

G(X)

is found by evaluating the probability

P(G(X) \leq u)

and showing it is uniform.

step 1

To find the tail distribution function of

Y = g(G(X))

, we calculate P(Y > y) for

y \in [0,1]

step 2

Since

g

is strictly increasing, P(Y > y) = P(g(G(X)) > y) = P(G(X) > g^{-1}(y))

step 3

Using the result from the previous question, P(G(X) > g^{-1}(y)) = 1 - P(G(X) \leq g^{-1}(y)) = g^{-1}(y)

step 4

The tail distribution function of

Y

g^{-1}(y)

Answer

The tail distribution function of

Y

g^{-1}(y)

Key Concept

Tail Distribution Function

Explanation

The tail distribution function of a transformed random variable

Y

is found by evaluating P(Y > y) using the inverse function

g^{-1}

step 1

To construct a random variable

Y

using

X, G^{-1}, g^{-1}

such that

Y

's tail distribution function is

g \circ G

, we define

Y = g(G(X))

step 2

From the previous steps, we know that

Y = g(G(X))

has a tail distribution function

g^{-1}(y)

step 3

To match the tail distribution function

g \circ G

, we use the fact that

g

is strictly increasing and

g(0) = 0, g(1) = 1

step 4

Therefore,

Y = g(G(X))

is the required construction

Answer

Y = g(G(X))

is the required construction.

Key Concept

Construction of a Random Variable

Explanation

The random variable

Y

is constructed using the composition of

g

and

G

applied to

X

to achieve the desired tail distribution function.