Question 1

Consider a dataset $X$ in $\mathbb{R}^3$. The dataset $X$ consists of 4 samples with 3 features each.The covariance matrix $C$ of this dataset has three non-zero eigenvalues which follow the given linear equations: $$\begin{aligned} 2\lambda_1 + 3\lambda_2 - \lambda_3 &= 4 \\ \lambda_1 - \lambda_2 + \lambda_3 &= 3 \\ \lambda_1 + \lambda_2 + 3\lambda_3 &= 15 \end{aligned}$$ Determine the variance of the given dataset. Consider a dataset of $n$ observations $\{x_1, x_2, ..., x_n\}$, where each $x_i$ follows a Bernoulli distribution with parameter $p$, i.e., $x_i \sim \text{Bernoulli}(p)$ for $i = 1, 2, ..., n$. However, you have reason to believe that the parameter $p$ might differ for two distinct groups within the dataset. You suspect that there are two groups in the dataset, each with its own parameter( $p_1$ and $p_2$). Now, develop an algorithm to estimate the parameters $p_1$ and $p_2$ using maximum likelihood estimation. Then, apply your algorithm to a dataset with the following observations and corresponding group labels: $\{0, 1, 1, 0, 1\}$ and $\{1, 0, 1, 1, 1\}$ for group 1 and group 2 respectively. Calculate the maximum likelihood estimates of $p_2$ and rounded to two decimal places. Decision tree diagram (x1 < 0.5, x2 > 0.5) with the question text