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

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 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.

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.

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 $p$.

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

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

The random variable X has $P(X=-1)=2p$, $P(X=0)=3p$, $P(X=1)=4p$, $P(X=2)=5p$. If the game is fair ($E[X]=0$), find $p$.

For a random variable X, $P(X=1)=p$, $P(X=2)=2p$, $P(X=3)=3p$, $P(X=4)=4p$. (a) Find $p$. (b) Compute $E[X]$.

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

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 $E[X]=1.5$, find $p$ and $q$.

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 $q$.