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

Given a discrete random variable X with $P(X=0)=0.2$, $P(X=1)=p$, and $P(X=2)=0.5$, find the value of $p$.

Let $X$ be a random variable with $P(X=0)=0.3$, $P(X=1)=0.4$, $P(X=2)=p$, and $P(X=3)=0.1$. Find $p$.

For a discrete random variable X, $P(X=1)=p$, $P(X=2)=2p$, $P(X=3)=3p$, and $P(X=4)=4p$. Determine $p$.

A random variable $X$ takes values 1, 2, 3 with probabilities $p$, $2p$, and $1-3p$, respectively. If $E(X)=2$, find $p$.

For a discrete variable $X$, $P(X=0)=p$, $P(X=1)=2p$, $P(X=2)=3p$, and $P(X=3)=4p$. (a) Find $p$. (b) Compute $E(X)$.

Consider a geometric distribution on $k=0,1,2,\dots$ with $P(X=k)=a\,p^k$. (a) Find $a$ in terms of $p$. (b) Compute $E(X)$.

Let $X$ satisfy $P(X=k)=x\,k$ for $k=1,2,3,4,5$. Given $P(X=1)=x$, find $x$ and then calculate $P(X>3)$.

A random variable $Y$ has $P(Y=k)=C\,k$ for $k=1,2,3$. (a) Determine $C$. (b) Find $ext{Var}(Y)$.

A random variable has $P(X=-1)=0.1$, $P(X=0)=p$, $P(X=1)=2p$, $P(X=2)=0.3$, and $P(X=3)=q$. Find $p$ and $q$.

A Poisson random variable $X$ has parameter $\lambda$. Prove that $\sum_{k=0}^\infty P(X=k)=1$ using the series expansion of $e^\lambda$.

Show that for a geometric distribution $P(X=k)=(1-p)p^k$ ($k=0,1,\dots$) the probabilities sum to 1 and derive $E(X)=p/(1-p)$.

In a binomial model $X\sim B(5,p)$ it is known that $P(X=2)=0.3$. Find $p$.