Before applying any algorithm to a real$-$world problem, it is common practice to first understand its behavior on synthetic data. Let$$s generate a synthetic data set for a regression problem: given input$-$output pairs, we aim to learn a function that maps inputs to outputs.

Suppose we have input

and output

that are related by the equation

$.$

However, in practice, we can only observe noisy data. That is$,$ given input

$,$ we observe output

that is related to

by the equation

$,$ where

is a random noise term. A common model for

is a Gaussian random variable with mean $0$ and variance

$.$

Suppose

$.$ Generate $10$ data points for

uniformly randomly distributed in the range of

$.$ Make a plot that contain the following elements:

$(1)$ The ground truth function.

$(2)$ The noisy data points.

$(3)$ Add a legend to the plot, where the ground truth is labeled as $$Ground truth$$ and the noisy data points are labeled as $$Noisy data$.$ Label the x$-$axis as $$x$$ and the y$-$axis as $$y$.$

A sample figure is shown below.

# code here

$../\_$images$/$a$3011438376$c$90858$f$866272$f$856$ad$29$e$463$b$2$dbf$1$ef$86$e$4999129$fa$739$fae$47.$png

Q$2$

We usually denote a normal distribution with mean

and variance

as

$.$

Let$$s generate a synthetic data set for a classification problem. Let X be a random variable that follows a normal distribution

and Y be a random variable that follows a normal distribution

$.$ X and Y can be some feature of two groups. For example, the height of high school students and the height of college students. Different groups can have different distributions of the same feature.

Generate $1000$ samples for X and $500$ samples for Y$.$ This models the scenario where we have more data for one group than the other.

$(1)$ Plot the histograms of X and Y in the same figure.

$(2)$ Add label $$X samples$$ to the histogram of X and label $$Y samples$$ to the histogram of Y$.$ Add a legend to the plot.

$(3)$ The two histograms should have different color and set the transparency to $0\mathrm{.}5$ so that we can see the overlap of the two histograms.

Hint: the transparency is usually named as alpha in most plotting libraries.

A sample figure is shown below.

# code here

$../\_$images$/54$d$9$c$5$e$95$db$5$a$97$e$323$e$3$c$204399$d$24$d$054405$d$09$aa$9$d$5$e$42$b$31$d$9$e$4$ba$1$fabef.png

Q$3.$ Let$$s bring our data science skill to the Wall Street. One model of stock price is the random walk model:

Suppose

is the stock price at day

$.$

is the initial stock price. At each day, the change of stock price is a random variable

$,$ which is normally distributed with mean

and variance

$.$ The stock price at day

is

$.$

$(1)$ Write a function stock$\_$price$\_$simulation, that take input

X$0$: the initial stock price

mu: the mean of the normal distribution

sigma: the standard deviation of the normal distribution

n: the number of days

Return a list $($or numpy array$)$ of stock prices at each day

$.$

# code here

$(2)$ Take

$.$ Sample $10$ trajectories of the stock price and plot them in the same graph.

A sample figure is shown below.

# code here

$../\_$images$/4$d$3532783$f$6$bbe$597$cb$4$c$1585510$a$55$c$09$c$4$ae$61867$a$938093$bafac$287$afb$5$d$9.$png

$(3)$ Estimate the expectation and standard deviation of the stock price on day $100,$ using $1000$ samples.

# code here

$(4)($Challenge$,$ not graded$)$ A call option is a contract that allows you to buy a stock at a fixed price at a future date. Suppose you own a call option that allows you to buy a stock at day $100$ at price $105($this is called the strike price$).$

If the stock price at day $100$ is above $105.$ Then you can exercise the option, pay $105$ to get the stock, and sell it at the market price to make a profit. Otherwise, you don$$t exercise the option and don$$t make a profit.

Estimate the probability that you can make a profit using the call option. Suppose you$$re the seller or the buyer of this call option. Estimate what should be the fair price of the call option.

# code here

Q$4$

One model of wealth inequality is the pareto distribution.

Let$$s generate N $=1000$ samples from a pareto distribution with parameters a$=20$ using the following code:

import numpy as np

N $=1000$

a $=20$

x $=$ np$.$random.pareto$($a$,$ N$)$

You can think of x as samples of wealth of a population.

$(1)$ Plot the histogram of the samples.

# code here

$(2)$ The k$-$quantile of a distribution is the value such that k$\%$ of the samples are less than or equal to the value. For example, the $50-$quantile is the median.

What are the median and the mean of the samples? What is the percentage of the population that are above$-$average wealthy?

# code here

$(3)$ Estimate what percentage of the population owns more than $80\%$ of the wealth?

Hint: you can sort the array such that

$,$ and compute the cumulative sum of the array:

$.$ Then

is the total wealth of the top i people