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

Let $X$ take values $k=1, 2, 3, 4, 5$ with $P(X=k)=pk$. Find $p$.

Let $X$ take values $0, 1, 2$ with probabilities $P(X=0)=p$, $P(X=1)=2p$, $P(X=2)=1-3p$. Determine all $p$ for which this is a valid distribution.

A loaded six-sided die satisfies $P(i)=\frac{i}{k}$ for $i=1,2,3,4,5,6$.

Find $k$.

The probability distribution of a discrete random variable $X$ is given by $P(X=-1)=2p$, $P(X=0)=3p$, $P(X=1)=4p$, and $P(X=2)=5p$.

Determine whether there is a value of $p$ for which the game is fair (i.e. $E[X]=0$).

A random variable $X$ has probabilities $P(X=0)=p$, $P(X=1)=2p$, $P(X=2)=3p$ and $P(X=3)=1-6p$. Determine the range of $p$ for which this is a valid distribution.

For a random variable $X$, $P(X=0)=p$, $P(X=1)=p$, $P(X=2)=0.3$, $P(X=3)=0.2$. Determine $p$ so that this is a valid probability distribution.

A game yields outcomes $X=-2, 1, 4$ with probabilities $0.1, p, 0.9-p$. Find $p$ so that the game is fair ($\text{E}[X]=0$).

Given $P(X=1) = 0.2$, $P(X=2) = p$, $P(X=3) = q$, $P(X=4) = 0.4$ and the mean $E[X] = 2.8$, find $p$ and $q$.

Given $P(X=0)=p$, $P(X=1)=2p$, $P(X=2)=q$, $P(X=3)=0.4$, and $E[X]=1.8$, find $p$ and $q$.

Given a discrete random variable $X$ with probabilities $P(X=1)=2p$, $P(X=2)=3p$ and $P(X=3)=0.2$, find the value of $p$.

The random variable $X$ has probabilities $P(X=-2)=p$, $P(X=-1)=0.1$, $P(X=1)=q$, and $P(X=2)=0.2$. If $X$ represents a fair game (i.e. $E(X)=0$) and the probabilities sum to 1, find $p$ and $q$.