Plotting 1: Scatterplots with ggplot

Use the ChickWeight dataset provided as default in R. The data describes the weight of chicks in four different diet groups over a number of days. Generate $4$ plots in ggplot shown in a $2\times 2$ layout of weight versus time of the chicks in each diet group. Also, in one long string of pipes, determine the average weight of chicks by diet group on the ${20}^{th}$ day of the study. Give your plots informative titles and labels.

Plotting 2: Histograms with ggplot

Use the CO2 dataset provided as default in R. The data describes carbon dioxide concentration and uptake values for a certain type of grass species.

Create a $1$ row by $2$ column set of ggplots. On the left panel, include only data from plants from Quebec; on the right, only from Mississippi. In each plot, show a boxplot of carbon dioxide uptake rates for each of the two treatments: Chilled and Non-Chilled. Fill in the boxplot with colors: one color for Non-Chilled, for both origins, another color for Chilled, for both origins. Give your plots informative titles and labels.

Function 1: Mixture Normal Distribution.

Write an R function to do the following. Generate n observations from a mixture of two normal distributions, i.e., generate U from a Bernoulli($p$) distribution and if $U=1$, then draw $Y$ from a normal distribution with mean ${\mu }_1$ and standard deviation ${\sigma }_1$, or if $U=0$, draw Y from the other normal distribution with mean ${\mu }_2$ and standard deviation ${\sigma }_2$.

In this context, there is a probability ($p$) that $Y$ is drawn from a normal distribution $N({\mu }_1,{\sigma }_1)$, and there is a probability $1-p$ that $Y$ is drawn from the other normal distribution $N({\mu }_2,{\sigma }_2)$. Note that the density of $Y$ is the sum of weighted density of two normal distributions.

Write a function with five arguments $(p,{\mu }_1,{\mu }_2,{\sigma }_1,{\sigma }_2)$ to sample $Y$ from the mixture normal distribution.

Draw $1000$ samples from mixture normal $(0.5,-2,3,2,1.5)$ and store in vector $Y1$ and draw $1000$ samples from mixture normal $(0.2,-2,3,2,1.5)$ and store in vector $Y2$. Set a seed for reproducibility.

Generate a two-panel plot in ggplot with the left panel containing the histogram and a fitted density for $Y1$ and the right panel containing the same for $Y2$.

Function 2: Weather Simulation.

Write a function to simulate the weather forecast in Richmond. Assume there are two states of weather in Richmond: sunny and rainy. If a day is sunny, the probability that the next day is sunny is $0.85$. If a day is rainy, the probability that the next day is rainy is $0.35$. If a day is rainy, the amount of rainfall accumulation in the city is governed by an $\text{Exponential}(\lambda =2)$ distribution, where the value from that distribution is the rainfall in inches. If a day is sunny, there can be no rain.

Specifically, given an initial day’s weather conditions, simulate the following $10$ days of weather and calculate the projected rainfall accumulation in inches.

• Write a function describing this process.
• First, assume sunny conditions on the initial day. Output the number of projected sunny days in the next $10$ days, as well as the projected rainfall accumulation. Repeat this $1000$ times and return the average of all these simulations.
• Now, assume rainy conditions on the initial day. Output the number of projected sunny days in the next $10$ days, as well as the projected rainfall accumulation. Repeat this $1000$ times and return the average of all these simulations.

Grading

A general rubric is below, in terms of points:

Plotting 1: 10 points
Plotting 2: 15 points
Function 1: 30 points
Function 2: 30 points
Package on Github: 10 points
Code is clear and commented: 5 points
Total: 100 Points