本次美国代写是机器学习数学理论的一个Problem Set

1 Basic Calculus [10 pts]

The following questions test your basic skills in computing the derivatives of univariate functions,

as well as applying the concept of convexity to determine the properties of the functions.

(a) (3 pts) Find all extrema of the function f(x) = ln(2 x2). For each extremum, state if it is

a maximum or a minimum.

(b) (3 pts) Show that f(x) = ln 1

1+e x is concave.

(c) (4 pts) Show that f(x) = e x2

is neither convex nor concave.

2 Continuous Random Variables [10 pts]

(a) (2 pts) Given a continuous random variable X with probability density function f(X), what

are the expressions for the mean and variance of this variable?

(b) (2 pts) Can the value of the probability density function (PDF) f(X) exceed 1? Why or why

not?

(c) (2 pts) Consider a random variable X that follows the uniform distribution between a and b,

i.e. its PDF is equal to a constant c on this interval, and 0 otherwise. Derive c in terms of a

and b.

(d) (2 pts) Derive the expected value of X in terms of a and b. Show all your steps.

(e) (2 pts) Derive the cumulative distribution function F(X) on the interval a X b.

3 Discrete Random Variables [10 pts]

(a) (2 pts) Two students taking a Machine Learning class became project partners. They are

trying to decide what operating system to use for the project. Suppose each student has a

laptop, which could be one of three types: Mac OS, Windows, or Linux. If the distribution of

laptops among students follows the PDF shown below, what is the probability that the two

teammates have dierent laptops?

Mac OS 0.6

Windows 0.3

Linux 0.1

Suppose we have three discrete random variables x, y and z that take values 0 or 1 according

to the distribution below.

x = 0

z = 0 z = 1

y = 0 0 1

12

y = 1 1

4

1

4

x = 1

z = 0 z = 1

y = 0 1

12

1

12

y = 1 0 1

4

(b) (2 pts) Find the joint distribution of y and z

(c) (2 pts) Find the marginal distributions of y and z

(d) (2 pts) Find the conditional distribution of x given that y = 0.

(e) (2 pts) Are y and z independent? Explain.

**程序辅导定制C/C++/JAVA/安卓/PYTHON/留学生/PHP/APP开发/MATLAB**

本网站支持 Alipay WeChatPay PayPal等支付方式

**E-mail:** vipdue@outlook.com **微信号:**vipnxx

如果您使用手机请先保存二维码，微信识别。如果用电脑，直接掏出手机果断扫描。