Question Type 3: Finding certain values of parameters such that the probability table holds true Exercises

The random variable $X$ takes the value $2$ with probability $p$ and $-3$ with probability $1-p$. Find $p$ such that the expected value is zero.

The random variable $X$ takes values $1$ with probability $p$ and $-1$ with probability $1-p$. Find $p$ so that the game is fair (i.e. $E[X]=0$).

The random variable $X$ has the distribution
$\quad P(X=0)=p, \\ \quad P(X=1)=2p,\\ \quad P(X=2)=1-3p.$
Find $p$ so that this is a valid probability distribution.

Let $X$ have
$\quad P(X=-1)=p, \\ \quad P(X=0)=2p, \\ \quad P(X=2)=1-3p.$
Find $p$ such that $P$ is a valid distribution.

The random variable $X$ takes values $-2$, $1$, $3$ with probabilities $p$, $q$, and $1-p-q$, respectively. Find $p$ and $q$ so that the game is fair ($E[X]=0$) and the probabilities sum to 1.

The random variable $X$ takes values $-2,-1,0,1,2$ with probabilities $P(X=-2)=p, \quad P(X=-1)=2p, \quad P(X=0)=1-6p, \quad P(X=1)=2p, \quad P(X=2)=p.$
Find the range of $p$ for which this is a valid probability distribution.

The random variable $X$ takes values $0$, $3$, and $5$ with probabilities $p$, $q$, and $1-p-q$, respectively. Find $p$ and $q$ such that $E[X]=4$ and the probabilities sum to 1.

Let $X$ have $P(X=-2)=p, \quad P(X=0)=2q, \quad P(X=5)=1-p-2q.$
Find $p$ and $q$ so that $E[X]=1$ and the probabilities sum to 1.

The variable $X$ takes values $1,2,3,4$ with probabilities $P(X=1)=p, \quad P(X=2)=2p, \quad P(X=3)=3p, \quad P(X=4)=1-6p.$
Find $p$ such that $E[X]=3$ and the distribution is valid.

The distribution of $X$ is $P(X=-1)=2p, \quad P(X=2)=3q, \quad P(X=4)=1-2p-3q.$Find $p$ and $q$ such that the game is fair ($E[X]=0$) and probabilities sum to 1.

$X$ takes four values $-2,0,1,3$ with probabilities
$\quad P(X=-2)=p, \\ \quad P(X=0)=q, \\ \quad P(X=1)=2p, \\ \quad P(X=3)=1-p-q-2p.$
Find $p,q$ so that the game is fair ($E[X]=0$) and probabilities sum to 1.