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

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

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

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

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

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

A website visitor has a 10% chance of making a purchase. Out of 500 visitors, let $X$ be the number of purchasers. Compute $E(X)$ and $\mathrm{Var}(X)$.

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

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

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

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