Statistics & Probability Probability Distributions Syllabus

Probability Distributions syllabus — full chapter and topic list

Expand any chapter to see its topics and sub-topics. This is the whole examinable outline for Probability Distributions in Statistics & Probability, not a summary of it.

1. 1

   Discrete Distributions

   5 topics

   • Bernoulli Distribution
   • Binomial Distribution
   • Poisson Distribution
   • Geometric and Negative Binomial
   • Hypergeometric Distribution
2. 2

   Continuous Distributions

   4 topics

   • Uniform Distribution
   • Exponential Distribution
   • Gamma and Beta Distributions
   • Weibull Distribution
3. 3

   The Normal Distribution

   4 topics

   • Properties of the Normal Curve
   • Standard Normal and Z-Tables
   • Empirical Rule (68-95-99.7)
   • Assessing Normality
4. 4

   Sampling Distributions

   4 topics

   • Distribution of the Sample Mean
   • Central Limit Theorem
   • Standard Error
   • Distribution of the Sample Proportion
5. 5

   Distributions for Inference

   4 topics

   • Student's t-Distribution
   • Chi-Square Distribution
   • F-Distribution
   • Degrees of Freedom

Probability Distributions flashcards for Statistics & Probability

20 of 50 cards from the Probability Distributions deck — real questions with worked answers.

1. What defines a Bernoulli distribution, and what are its parameter and support?

   A Bernoulli distribution models a single trial with two outcomes (success/failure). Parameter $p$ = probability of success, support $x \in \{0,1\}$, with pmf $P(X=x)=p^{x}(1-p)^{1-x}$.

2. State the mean and variance of a Bernoulli$(p)$ random variable.

   $E[X]=p$ and $\operatorname{Var}(X)=p(1-p)$.

3. What does a Binomial distribution model, and what are its parameters?

   It models the number of successes in $n$ independent Bernoulli trials each with success probability $p$. Parameters: $n$ (number of trials) and $p$ (success probability).

4. Write the pmf of a Binomial$(n,p)$ random variable.

   $$P(X=k)=\binom{n}{k}p^{k}(1-p)^{n-k},\quad k=0,1,\dots,n.$$

5. Give the mean and variance of a Binomial$(n,p)$ distribution.

   $E[X]=np$ and $\operatorname{Var}(X)=np(1-p)$.

6. What does the Poisson distribution model, and what is its single parameter?

   It models the number of events occurring in a fixed interval of time or space when events happen independently at a constant average rate. Parameter $\lambda$ = mean number of events.

7. Write the pmf of a Poisson$(\lambda)$ random variable.

   $$P(X=k)=\frac{e^{-\lambda}\lambda^{k}}{k!},\quad k=0,1,2,\dots$$

8. What is the notable relationship between the mean and variance of a Poisson distribution?

   They are equal: $E[X]=\operatorname{Var}(X)=\lambda$.

9. Under what conditions can a Binomial distribution be approximated by a Poisson distribution?

   When $n$ is large and $p$ is small, with $\lambda=np$ moderate. Then Binomial$(n,p)\approx$ Poisson$(np)$.

10. What does the Geometric distribution model, and give its pmf (trials until first success).

    It models the number of trials until the first success. With success probability $p$: $$P(X=k)=(1-p)^{k-1}p,\quad k=1,2,3,\dots$$

11. State the mean and variance of a Geometric distribution (support $k\ge 1$).

    $E[X]=\dfrac{1}{p}$ and $\operatorname{Var}(X)=\dfrac{1-p}{p^{2}}$.

12. What memoryless property does the Geometric distribution possess?

    $P(X>m+n \mid X>m)=P(X>n)$: past failures do not affect the future number of trials needed for a success.

13. What does the Negative Binomial distribution model?

    It models the number of trials (or failures) needed to achieve a fixed number $r$ of successes, each with probability $p$. It generalizes the Geometric ($r=1$).

14. Give the pmf of the Negative Binomial distribution (number of failures $k$ before the $r$-th success).

    $$P(X=k)=\binom{k+r-1}{k}p^{r}(1-p)^{k},\quad k=0,1,2,\dots$$

15. What does the Hypergeometric distribution model, and how does it differ from the Binomial?

    It models successes in $n$ draws without replacement from a finite population of size $N$ containing $K$ successes. Unlike the Binomial, draws are not independent (no replacement).

16. Write the pmf of the Hypergeometric distribution with population $N$, successes $K$, and $n$ draws.

    $$P(X=k)=\frac{\binom{K}{k}\binom{N-K}{n-k}}{\binom{N}{n}}.$$

17. State the mean of a Hypergeometric distribution.

    $E[X]=n\dfrac{K}{N}$ (analogous to $np$ with $p=K/N$).

18. Define the continuous Uniform distribution on $[a,b]$ and give its pdf.

    All values in $[a,b]$ are equally likely. pdf: $$f(x)=\frac{1}{b-a},\quad a\le x\le b,$$ and $0$ otherwise.

19. State the mean and variance of a continuous Uniform$(a,b)$ distribution.

    $E[X]=\dfrac{a+b}{2}$ and $\operatorname{Var}(X)=\dfrac{(b-a)^{2}}{12}$.

20. What does the Exponential distribution model, and give its pdf with rate $\lambda$.

    It models waiting time between events in a Poisson process. pdf: $$f(x)=\lambda e^{-\lambda x},\quad x\ge 0.$$

See more Probability Distributions flashcards →

Planning Probability Distributions for Statistics & Probability

Probability Distributions (Statistics & Probability) FAQ