Cards In This Deck

• $\text{First step to find the value of a constant }k\text{ in a probability distribution function }P(X=x)=kx\text{?}$
  $$\begin{gathered} \text{Set the sum of all probabilities equal to 1.} \\ \sum P(X=x)=1 \end{gathered}$$
• $$\begin{gathered} \text{True or False?} \\ \text{The height of a randomly selected student is a discrete random variable.} \end{gathered}$$
  $$\begin{gathered} \text{❌ False} \\ \text{Height is measured on a continuous scale, so it is a continuous random variable. Discrete variables usually count distinct items (e.g., number of heads).} \end{gathered}$$
• $\text{First step to solve for two unknowns (}a\text{ and }b\text{) in a probability table, given }E(X)\text{?}$
  $$\begin{gathered} \text{Form two simultaneous equations:} \\ \text{1. }\sum P(X=x)=1 \\ \text{2. }\sum xP(X=x)=E(X) \end{gathered}$$
• $$\begin{gathered} \text{True or False?} \\ \text{If }X\text{ is a discrete random variable taking integer values }1,2,3\text{, then }F(2.6)\text{ is undefined or equal to 0.} \end{gathered}$$
  $$\begin{gathered} \text{❌ False} \\ F(2.6)=P(X\le 2.6)\text{. Since }X\text{ only takes integer values, this is equal to }P(X\le 2)\text{ or }F(2)\text{.} \end{gathered}$$
• $$\begin{gathered} \text{True or False?} \\ \text{For a cumulative distribution function }F(x)\text{, the value of }F(x)\text{ for the largest possible value of }x\text{ must be 1.} \end{gathered}$$
  $\text{✅ True}$
• $$\begin{gathered} \text{True or False?} \\ \text{If }E(X)=3.5\text{ for a roll of a die, it means the most likely outcome is 3.5.} \end{gathered}$$
  $$\begin{gathered} \text{❌ False} \\ \text{It is impossible to roll a 3.5. The Expected Value }E(X)\text{ is the theoretical long-run average.} \end{gathered}$$
• $\text{Which formula is generally quicker to calculate }Var(X)\text{ given a frequency table or probability distribution?}$
  $Var(X)=E(X^2)-(E(X){)}^2$
• $$\begin{gathered} \text{True or False?} \\ E(X^2)=(E(X){)}^2 \end{gathered}$$
  $$\begin{gathered} \text{❌ False} \\ E(X^2)=\sum x^{2}P(X=x) \\ (E(X){)}^2\text{ is the square of the mean. They are generally not equal.} \end{gathered}$$
• $\text{How do you calculate }Var(X)\text{ for a Discrete Random Variable given the probability distribution?}$
  $$\begin{gathered} \text{1. Calculate }E(X)=\sum xP(X=x) \\ \text{2. Calculate }E(X^2)=\sum x^{2}P(X=x) \\ \text{3. Use formula: }Var(X)=E(X^2)-[E(X){]}^2 \end{gathered}$$
• $\text{Formula for }E(aX+b)\text{?}$
  $aE(X)+b$
• $$\begin{gathered} \text{True or False?} \\ Var(3X-2)=3Var(X)-2 \end{gathered}$$
  $$\begin{gathered} \text{❌ False} \\ \text{✅ Correct Answer: }{3}^{2}Var(X)=9Var(X) \end{gathered}$$
• $\text{For a Discrete Random Variable, what is the formula for }Var(aX+b)\text{?}$
  $Var(aX+b)=a^{2}Var(X)$
• $\text{Condition to use the formula }E(X)=\frac{n+1}{2}\text{ for a Discrete Uniform Distribution?}$
  $\text{The random variable }X\text{ must take values over the set }\{1,2,3,...,n\}\text{.}$
• $\text{When modelling an experiment where there are }n\text{ distinct outcomes and each is equally likely, which distribution should be used?}$
  $\text{Discrete Uniform Distribution}$