Question Type 1: Finding the probability of certain outcomes of random variables Exercises

A random variable $X$ has distribution $P(X=0)=0.1$, $P(X=1)=0.2$, $P(X=2)=0.3$, $P(X=3)=0.4$. Find $P(X\le2)$.

A random variable $X$ has pmf $P(X=0)=0.3$, $P(X=1)=0.4$, $P(X=2)=0.3$. Find $E(X)$.

Let $X$ be a discrete random variable with probability mass function $P(X=x)=\frac{6}{11}\frac{1}{x}$ for $x=1,2,3$. Find $P(X>1)$.

Let $X$ be a discrete random variable with pmf $P(X=x)=\frac{c}{2^x}$ for $x=1,2,3,4$. Determine the constant $c$.

For $X$ with pmf $P(X=x)=0.5^x(1-0.5)$ for $x=0,1,2,\dots$, calculate $P(X\le1)$.

Let $X$ be a discrete random variable with pmf $P(X=x)=kx^2$ for $x=1,2,3,4$. Determine the constant $k$.

Let $X$ be a discrete random variable with $P(X=1)=p, P(X=2)=2p, P(X=3)=1-3p$. Determine all values of $p$ for which this is a valid probability distribution.

A random variable $X$ has pmf $P(X=x)=\frac{x+1}{15}$ for $x=0,1,2,3,4$. Compute $E(X)$.

Given pmf $P(X=0)=q, P(X=1)=2q, P(X=2)=3q, P(X=3)=4q$, find $q$ and then compute $E(X)$.

A loaded six‐sided die has probabilities $P(X=i)=\frac{i}{21}$ for $i=1,2,3,4,5,6$. Find $E(X)$.

Let $X$ take values $-1,2,5$ with probabilities $P(X=-1)=p, P(X=2)=q, P(X=5)=1-p-q$. Determine $p$ and $q$ so that the game is fair (i.e. $E(X)=0$).

A spinner has sides 1,2,3,4 with probabilities $P(1)=p$, $P(2)=0.1$, $P(3)=0.3$, $P(4)=q$. A play costs $2 and you win the number of dollars shown. Find $p$ and $q$ so that the game is fair.