Sixth Term Examination Paper (STEP) Probability and Statistics Syllabus

Probability and Statistics syllabus — full chapter and topic list

Expand any chapter to see its topics and sub-topics. This is the whole examinable outline for Probability and Statistics in Sixth Term Examination Paper (STEP), not a summary of it.

1. 1

   Probability

   6 topics

   • Sample spaces, events and the axioms of probability
   • Conditional probability and the multiplication rule
   • Independence and mutually exclusive events
   • Tree diagrams, Venn diagrams and the addition rule
   • Bayes' theorem and the total probability formula
   • Permutations, combinations and counting arguments
2. 2

   Discrete Random Variables

   6 topics

   • Probability distributions and cumulative distributions
   • Expectation, variance and their algebra
     • E(aX+b) and Var(aX+b)
     • Expectation of a function of X
   • The binomial distribution and its conditions
   • The Poisson distribution and approximation to the binomial
   • The geometric and discrete uniform distributions
   • Probability and moment generating functions
3. 3

   Continuous Random Variables

   5 topics

   • Probability density and cumulative distribution functions
   • Expectation, variance, median and mode for continuous variables
   • The continuous uniform and exponential distributions
   • The normal distribution and standardisation
     • Normal approximation to binomial and Poisson
     • Continuity corrections
   • Distributions of functions of a continuous variable
4. 4

   Data, Estimation and Inference

   5 topics

   • Measures of location and spread; interpreting data
   • Sampling, estimators and the distribution of the sample mean
     • Unbiased estimators
     • Central limit theorem (informal use)
   • Confidence intervals for a mean
   • Hypothesis testing: null/alternative, significance and errors
     • One- and two-tailed tests
     • Type I and Type II errors
   • Correlation, regression and the product moment coefficient

Probability and Statistics flashcards for Sixth Term Examination Paper (STEP)

18 of 51 cards from the Probability and Statistics deck — real questions with worked answers.

1. State the three Kolmogorov axioms of probability for events in a sample space $S$.

   For any event $A$, $P(A)\geq 0$ (non-negativity). For the whole sample space, $P(S)=1$ (normalisation). For mutually exclusive events $A_1,A_2,\dots$, $P\!\left(\bigcup_i A_i\right)=\sum_i P(A_i)$ (countable additivity).

2. Define a sample space, an event, and an elementary (simple) event.

   The sample space $S$ is the set of all possible outcomes of an experiment. An event is any subset $A\subseteq S$. An elementary event is a single-outcome subset $\{\omega\}$ containing just one element of $S$.

3. What is the complement rule, and what is $P(\varnothing)$?

   $P(A')=1-P(A)$, where $A'$ is the complement of $A$. The impossible event satisfies $P(\varnothing)=0$.

4. Define the conditional probability $P(A\mid B)$ and state the condition for it to be defined.

   $P(A\mid B)=\dfrac{P(A\cap B)}{P(B)}$, provided $P(B)>0$. It is the probability of $A$ given that $B$ has occurred.

5. State the multiplication (chain) rule for $P(A\cap B)$ and its extension to three events.

   $P(A\cap B)=P(A\mid B)\,P(B)=P(B\mid A)\,P(A)$. For three events, $P(A\cap B\cap C)=P(A)\,P(B\mid A)\,P(C\mid A\cap B)$.

6. Define independence of two events $A$ and $B$, and give the equivalent conditional form.

   $A$ and $B$ are independent if $P(A\cap B)=P(A)\,P(B)$. Equivalently, $P(A\mid B)=P(A)$ (when $P(B)>0$): knowing $B$ does not change the probability of $A$.

7. What is the difference between mutually exclusive events and independent events?

   Mutually exclusive means $A\cap B=\varnothing$ so $P(A\cap B)=0$ — they cannot both occur. Independent means $P(A\cap B)=P(A)P(B)$. For events with positive probability the two properties are incompatible: mutually exclusive events are dependent, since one occurring forces the other not to.

8. State the addition rule (inclusion–exclusion) for two events and for three events.

   $P(A\cup B)=P(A)+P(B)-P(A\cap B)$. For three events, $P(A\cup B\cup C)=P(A)+P(B)+P(C)-P(A\cap B)-P(A\cap C)-P(B\cap C)+P(A\cap B\cap C)$.

9. In a tree diagram, how do you obtain the probability of a particular path and the probability of an end event?

   Multiply the conditional probabilities along the branches of a single path. To get the probability of an overall outcome, add the probabilities of all distinct paths (mutually exclusive routes) that lead to it.

10. State the law of total probability for a partition $B_1,\dots,B_n$ of the sample space.

    $P(A)=\sum_{i=1}^{n} P(A\mid B_i)\,P(B_i)$, where the $B_i$ are mutually exclusive and exhaustive with $P(B_i)>0$.

11. State Bayes' theorem in its standard form for events $A$ and $B$.

    $P(A\mid B)=\dfrac{P(B\mid A)\,P(A)}{P(B)}=\dfrac{P(B\mid A)\,P(A)}{\sum_i P(B\mid A_i)\,P(A_i)}$, where the denominator uses the total probability formula over a partition $\{A_i\}$.

12. Define the number of permutations of $r$ objects chosen from $n$ distinct objects.

    $^nP_r=\dfrac{n!}{(n-r)!}=n(n-1)\cdots(n-r+1)$. Order matters and there is no repetition.

13. Define the binomial coefficient (number of combinations) $\binom{n}{r}$ and give its symmetry property.

    $\binom{n}{r}=\dfrac{n!}{r!\,(n-r)!}$, the number of ways to choose $r$ objects from $n$ without regard to order. Symmetry: $\binom{n}{r}=\binom{n}{n-r}$.

14. How many distinct arrangements are there of $n$ objects of which $n_1,n_2,\dots,n_k$ are identical within each group?

    $\dfrac{n!}{n_1!\,n_2!\cdots n_k!}$, the multinomial coefficient, where $n_1+n_2+\cdots+n_k=n$.

15. For a discrete random variable, define the probability mass function and the conditions it must satisfy.

    The pmf is $p(x)=P(X=x)$. It must satisfy $p(x)\geq 0$ for all $x$, and $\sum_x p(x)=1$ over all values $x$ in the range of $X$.

16. Define the cumulative distribution function $F(x)$ and state its key properties.

    $F(x)=P(X\leq x)$. It is non-decreasing, right-continuous, with $\lim_{x\to-\infty}F(x)=0$ and $\lim_{x\to\infty}F(x)=1$. For discrete $X$, $F$ is a step function and $P(a<X\leq b)=F(b)-F(a)$.

17. Define the expectation $E(X)$ of a discrete random variable and $E(g(X))$.

    $E(X)=\sum_x x\,p(x)$. For a function, $E(g(X))=\sum_x g(x)\,p(x)$ (the law of the unconscious statistician).

18. Define the variance of $X$ and give the computational formula.

    $\operatorname{Var}(X)=E\!\big[(X-\mu)^2\big]$ where $\mu=E(X)$. Computationally, $\operatorname{Var}(X)=E(X^2)-\big(E(X)\big)^2$. The standard deviation is $\sigma=\sqrt{\operatorname{Var}(X)}$.

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