Expectation & variance

What the problems look like

A few statements from this skill. Answers, hints and worked solutions live in the practice session.

1. Warm-up

   A binary classifier predicts the positive class on any given input with probability $p = 1/5$. Model this as a Bernoulli random variable $X$, where $X=1$ for a positive prediction and $X=0$ otherwise. Find (i) $E[X]$ and (ii) $\operatorname{Var}(X)$, each as a fraction.

2. Warm-up

   A binary classifier predicts the positive class on any given input with probability $p = 1/4$. Model this as a Bernoulli random variable $X$, where $X=1$ for a positive prediction and $X=0$ otherwise. Find (i) $E[X]$ and (ii) $\operatorname{Var}(X)$, each as a fraction.

3. Warm-up

   A binary classifier predicts the positive class on any given input with probability $p = 1/6$. Model this as a Bernoulli random variable $X$, where $X=1$ for a positive prediction and $X=0$ otherwise. Find (i) $E[X]$ and (ii) $\operatorname{Var}(X)$, each as a fraction.

Problems shown under their original terms: Generated (template, solver-computed).