Question 1: If X \sim \text{Normal}(0, 1) and Y = X^2, then show that…

1. If $X \sim \text{Normal}(0, 1)$ and $Y = X^2$, then show that $Y \sim \chi^2_1$. Write all the steps clearly. [4 Marks]

2. Consider the following R code and answer the sub-questions based on the code.

(i) What does the command x <- rpois(N, lambda = 2) do? Explain your answer. [2 Marks]

(ii) Interpret the output of the command summary(x). [2 Marks]

(iii) What are the values of the theoretical mean and the observed sample mean of the distribution $X$? [1 Mark]

(iv) Are the theoretical mean and variance close to the sample mean and sample variance? Why or why not? Justify your answer. [2 Marks]

1. A nutrition researcher is investigating whether two different diets (Diet A and Diet B) affect people's daily protein intake differently. She records the protein intake (in grams) of individuals on the two diets, and the observations $y_{ij}$ is shown in the following table:

Using a one way classification model, answer the given subquestions.

(i) Calculate $\overline{Y}_{1o}, \overline{Y}_{2o}, \overline{Y}_{oo}$. [4 Marks]

(ii) Find the value of sum of squares within group (SSW). [4 Marks]

(iii) Find the value of sum of squares between group (SSB). [2 Marks]

(iv) Calculate the value of total sum of squares (TSS). [4 Marks]

(v) Verify that TSS $=$ SSB $+$ SSW. [1 Mark]

1. Suppose we have data $D = \{(x, y) : (0, 3), (1, 4), (2, 7)\}$ and assume the linear model $(\vec{y}, X\vec{\beta}, I_{3\times3})$ for the data.

(i) Write the linear regression model for the given data and then find $\vec{Y}, X, \vec{\beta}$ and $\vec{\epsilon}$. [2 Marks]

(ii) Find the least square estimate for $\vec{\beta}$. [5 Marks]

(iii) Write the fitted equation and estimate the value of $y$ when $x = 3$. [2 Marks]