東京大学 情報理工学研究科 2023年8月実施 数学 第2問

Author

zephyr

Description

Consider a function defined by the following integral for positive real numbers .

Answer the following questions. You may answer without showing that the above integral converges.

(1) Find the value of .

(2) The inequality holds for any positive real number and non-negative integer .

• (a) For positive real numbers , show the following inequality.

• (b) When , show that the following inequality holds for any real number that satisfies .

(3) When the second-order derivative of is expressed as

find a function . You may answer without showing that the order of differentiation and integration can be exchanged.

(4) Find the value of defined as

Here, you may use the fact that the following relation holds.

(5) Define a function for positive real numbers and as

Find the value of defined as

Kai

(1)

We start by evaluating :

This integral is well-known and is the Laplace transform of a constant function . Evaluating the integral:

So, .

(2)

Part 1: Show that for positive real numbers

To show this inequality, we start by considering the definition of :

The problem gives us the inequality for any positive real number and non-negative integer . Taking the reciprocal and considering the exponential function in the integrand:

Substituting this into the integral, we obtain:

The integral converges when . Evaluating this integral:

Thus, the inequality becomes:

Now, by setting , we obtain:

Therefore, we have shown that:

Part 2: Show that for and

We are given that and , and we need to prove the inequality:

Using the same inequality , we substitute it into the integral:

Next, we evaluate the integral:

Since , the term is less than , which means:

Thus:

(3)

Given:

We first need to compute the second derivative of :

First derivative:

Second derivative:

Thus, .

(4)

We need to find the value of the expression

This problem requires us to calculate two integrals: one involving and another involving . We are also given the hint that the following relation holds:

Step 1: Expressing the Second-Order Derivative of

From Question 3, we know that:

Setting :

This integral represents the first term in , which is:

Thus, we have:

Step 2: Calculating the First Integral

The value of the second derivative of the logarithm of at is given as:

We know that:

At , , is the first moment (which is ), and is the second moment (which is ).

We can express:

Given:

Thus:

(5)

We are asked to find the value of , defined as:

where the function is given by:

Step 1: Identify the form of

The function is a probability density function corresponding to a Rayleigh distribution, with the parameter . The Rayleigh distribution has the form:

which is commonly used to describe the distribution of the magnitude of a two-dimensional vector with independent and identically distributed normal components.

Step 2: Calculation of the first moment

We need to compute the expected value of under this distribution, given by:

We perform a substitution to simplify this integral:

Let , hence .

The integral becomes:

This simplifies to:

The first integral evaluates to 1 because it is the integral of the exponential distribution. The second integral is a well-known result:

where is the Euler-Mascheroni constant. Thus,

Step 3: Calculation of the second moment

Next, we need to compute the second moment:

Using the same substitution :

This expands to:

Using the known results:

and

we have:

Step 4: Calculate

Finally, is the variance, which is given by:

Substitute the values:

so:

Subtracting:

Simplifying:

This is the final value of .

Knowledge

Gamma函数 不定积分 定积分 方差

解题技巧和信息

1. Gamma Function: Recognize that represents the Gamma function .
2. Inequality Manipulation: Use known inequalities such as to estimate integrals.
3. Variance Calculation: The variance of logarithms of exponential and Rayleigh distributed variables often results in expressions involving .

重点词汇

• Gamma function 伽马函数
• Inequality 不等式
• Variance 方差
• Logarithm 对数
• Rayleigh distribution 瑞利分布
• Logarithm 对数
• Euler-Mascheroni constant 欧拉-马歇罗尼常数
• Variance 方差