IB Mathematics Applications & Interpretation Question Type 3: Finding Certain Values of Parameters Such That the Probability Table Holds True Exercises

The distribution of a discrete random variable $X$ is given by:

$$P(X=-1)=2p,P(X=2)=3q,P(X=4)=1-2p-3q$$

Find $p$ and $q$ such that the game is fair ($E[X]=0$) and all probabilities are valid.

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

Find an expression for $q$ in terms of $p$ such that $E[X]=1$, and determine the range of possible values for $p$.

A discrete random variable $X$ takes values $1,2,3,4$ with probabilities given by: $$\text{P}(X=1)=p$$ $$\text{P}(X=2)=2p$$ $$\text{P}(X=3)=3p$$ $$\text{P}(X=4)=1-6p$$

Find the value of $p$ such that $\text{E}[X]=3$, ensuring that the probability distribution is valid.

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

A discrete random variable $X$ takes values $0,3$, and $5$ with probabilities $p,q$, and $1-p-q$ respectively.

Given that $\text{E}[X]=4$, express $q$ in terms of $p$ and determine the range of possible values for $p$.

The random variable $X$ has the following probability distribution:

$$\begin{array}{rl}P(X=0)& =p\\ P(X=1)& =2p\\ P(X=2)& =1-3p\end{array}$$

Find the set of values of $p$ for which this is a valid probability distribution.

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.

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

$$\begin{array}{ccccc}x& -2& 0& 1& 3\\ P(X=x)& p& q& 2p& 1-3p-q\end{array}$$

Find an expression for $q$ in terms of $p$ such that the game is fair ($E[X]=0$), and determine the range of possible values for $p$.

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

Find $q$ in terms of $p$ such that $E[X]=0$, and determine the range of possible values for $p$.

Let the discrete random variable $X$ have the following probability distribution:

$$\begin{array}{rl}\text{P}(X=-1)& =p\\ \text{P}(X=0)& =2p\\ \text{P}(X=2)& =1-3p\end{array}$$

Find the range of possible values for $p$ such that this is a valid probability distribution.