Question Type 2: Finding the expected value and variance for a binomially distributed variable Exercises

Given $X \sim B(n, p)$ with $E(X) = 20$ and $p = 0.4$, determine $n$ and $\text{Var}(X)$.

A multiple-choice test has $10$ questions, each with four options. A student guesses randomly on each question. Let $X$ be the number of correct answers. Find the expected value and the variance of $X$.

A fair coin is tossed 20 times. Let $X$ be the number of heads. Find the expected value and variance of $X$.

A website visitor has a $10\%$ chance of making a purchase. Out of $500$ visitors, let $X$ be the number of purchasers.

Calculate $E(X)$ and $\mathrm{Var}(X)$.

In a production run of 50 items, each item has a 0.2 probability of being defective. Let $X$ be the number of defective items. Calculate $\text{E}(X)$ and $\text{Var}(X)$.

Suppose $X \sim B(n, p)$ and you know $E(X) = 16$ and $\text{Var}(X) = 12$. Find the values of $n$ and $p$.

A basketball player makes free throws with probability $0.8$. If she takes $40$ shots, let $X$ be the number she makes. Determine $\text{E}(X)$ and $\text{Var}(X)$.

A quality-control inspector checks 200 components. Each component has a 5% chance of being faulty. Let $X$ be the number of faulty components. Find the expected value and variance of $X$.

Out of $150$ electronic components, each fails independently with probability $0.02$. Let $X$ be the number of failures. Find $\mathrm{E}(X)$ and $\mathrm{Var}(X)$.

A biased die is rolled 60 times. Let $X$ be the number of times a six appears. Given that $P(\text{six}) = \frac{1}{6}$, find $\mathrm{E}(X)$ and $\mathrm{Var}(X)$.

A binomial random variable $X$ has parameters $n = 20$ and an unknown $p$. If $E(X) = 8$, find $p$ and then compute $\mathrm{Var}(X)$.

In a survey of 100 randomly selected people, each has probability 0.3 of owning a pet. Let $X$ be the number of pet owners. Find $E(X)$ and $\mathrm{Var}(X)$.