IB Mathematics Applications & Interpretation Question Type 2: Finding the Expected Value of Outcomes of Random Variables Exercises

A discrete random variable $X$ has the following probability distribution:

Given that $q=2p$, find the value of $p$ and the value of $q$.

Show that for a geometric distribution $P(X=k)=(1-p)p^k$ for $k=0,1,2,\dots$, where $0<p<1$, the probabilities sum to 1 and derive $E(X)=\frac{p}{1-p}$.

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 $p$.

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

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

Let $X$ be a discrete random variable such that $P(X=k)=xk$ for $k=1,2,3,4,5$.

Given that $P(X=1)=x$, find the value of $x$ and calculate $P(X>3)$.

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

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 $p$.

A discrete random variable $X$ takes values 1, 2, and 3 with probabilities $p$, $2p$, and $1-3p$, respectively.

Given that $\text{E}(X)=2$, find the value of $p$.