Question Type 2: Finding the expected value of outcomes of random variables Exercises

A game pays $n^2$ dollars where $n$ is the number obtained from rolling a fair four-sided die (values 1 to 4). Find expected payout $E(X)$.

A fair six-sided die is rolled once. Find the expected value of the outcome $X$.

A spinner is divided into three regions labeled 1, 2, and 3 with probabilities $0.2$, $0.5$, and $0.3$ respectively. Determine the expected value of the spinner outcome $X$.

A discrete random variable $X$ has pmf $P(X=k)=c/k^2$ for $k=1,2,3$, and $0$ otherwise. Find the constant $c$ and then compute $E(X)$.

A factory produces lightbulbs. Let $X$ be the lifetime (in thousands of hours) of a bulb, with pmf $P(X=1)=0.2$, $P(X=2)=0.5$, $P(X=3)=0.3$. Find $E(X)$.

An insurance company charges a premium of 1000 dollars. The loss $L$ in a year is $0$ with probability $0.9$, $5000$ with probability $0.08$, or $20000$ with probability $0.02$. Let profit $X = 1000 - L$. Compute $E(X)$.

A random variable $X$ has the probability mass function given by

Find $E(X)$.

A lottery ticket costs $2. The prize distribution is: win $50 with probability $0.01$, win $5 with probability $0.05$, or win nothing with probability $0.94$. Let $X$ be the net gain. Compute $E(X)$.

A random variable $X$ takes values $0,1,2,3,4$ with equal probability. Find the variance $\text{Var}(X)$ and remark on the relationship between $\text{Var}(X)$ and $\text{E}(X)$.

A box contains 5 red, 3 blue, and 2 green balls. You draw one ball at random. Let $X=1$ if you draw red, $X=2$ if blue, and $X=3$ if green. Find $E(X)$.

A biased coin gives a payoff of $5$ dollars for heads and $-3$ dollars for tails. The probability of heads is $0.4$. Find the expected payoff $E(X)$.