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  • STA256 Lecture 17

Special Distributions

• Identify correct distributions and apply
• Bernoulli Distribution
• Binomial Distribution
• Geometric Distribution
• Negative Binomial Distribution
• Hypergeometric Distribution
• Gamma Distribution
• Continuous Uniform Distribution
• Exponential Distribution
• Chi-squared Distribution
• Beta Distribution
• Normal Distribution
• Standard Normal Distribution
• Examples:
  • Example:

    • In mario kart, when a player is in first place, you have the probabilities for items from item boxes as:

    • $\begin{array}{cc}\text{Item}& \text{Probability}\\ \text{Banana}& 0.10\\ \text{Green Shell}& 0.05\\ \text{Coin}& 0.85\end{array}$

    • Consider a player who remains in first place

    • a:

      • What's the probability of the player obtaining a green shell on their $5^{\text{th}}$ item box, with any different items on the prior $4$ trials:
      • Failures from trial $1\to 4$ and a success on trial $5$
      • Geometric Distribution
        • $p=0.05$ by the table
        • $x=$ failure trials
        • $x+1=$ success trial
        • $P(X=x)=0.05(0.95{)}^4$
        • $P(X=x)\approx 0.0407$

    • b:

      • What's the probability of obtaining $2$ green shells in $5$ item boxes
      • Binomial Distribution
        • $n=5$
        • $p=0.05$
        • $P(X=x)=(\genfrac{}{}{0ex}{}{n}{x})p^x(1-p{)}^{n-x}$
          • Where $x=$ number of successes
        • $P(X=2)=(\genfrac{}{}{0ex}{}{5}{2}){0.05}^2(0.95{)}^{5-2}\approx 0.0214$

    • c:

      • What's the probability of obtaining the $2^{\text{nd}}$ green shell on the $5^{\text{th}}$ item box.
      • Negative Binomial Distribution
        • $r=$ the number of the success
        • $\text{failures}+\text{successes}=\text{trials}$
        • $\text{failures}+r=\text{trials}$
        • $3+2=5$
        • failures are $x$

        ---

        • $r=2$ is our second success
        • $5^{\text{th}}$ trial is $x+r=5$
        • so $x=3$ as failures
        • $P(X=x)=(\genfrac{}{}{0ex}{}{x+r-1}{r-1})p^r(1-p{)}^x$
        • $P(X=3)=(\genfrac{}{}{0ex}{}{4}{1}){0.05}^2(0.95{)}^3$
        • $P(X=3)\approx (\genfrac{}{}{0ex}{}{4}{1})0.0021434375$
        • $P(X=3)\approx 0.00857$
  • Example:

    • A class of $30$ has $10$ stats majors, $12$ cs majors and $8$ math majors

    • d:

      • $8$ students are randomly selected for a tournament
      • What is the probability that $2$ of them are stats majors?
      • Hypergeometric Distribution
      • $N=30$
      • $D=10$
      • $N-D=20$
      • $n=8$
      • $x=2$
        $n-x=6$
      • $P(X=x)=\frac{(\genfrac{}{}{0ex}{}{D}{x})(\genfrac{}{}{0ex}{}{N-D}{n-x})}{(\genfrac{}{}{0ex}{}{N}{n})}$
      • $P(X=2)=\frac{(\genfrac{}{}{0ex}{}{10}{2})(\genfrac{}{}{0ex}{}{20}{6})}{(\genfrac{}{}{0ex}{}{30}{8})}$
      • $P(X=2)=\frac{\frac{10!}{2!(10-2)!}\frac{20!}{6!(20-6)!}}{\frac{30!}{8!(30-8)!}}$
      • $\dots$
      • $P(X=2)\approx 0.2980$