Chapters 1–5

Formula and concept review

Data summaries, probability,
and discrete models

EN.553.211 · Probability & Statistics for the Life Sciences

Name the object → choose the method

Data, events, and random variables

Observed data

x_1,\ldots,x_n

An event

\textcolor{#8bded6}{A}\subseteq S

A random variable

\textcolor{#f4dc88}{X}:S\longrightarrow\mathbb{R}

Several outcomes can give one value

Meaning

Categorical variables label groups. Quantitative variables measure amounts. An event collects outcomes. A random variable assigns a number to each outcome.

When to use it

“Describe the data” calls for a statistic. “How likely?” calls for an event. “Number of…” often defines a count X.

Population:

\mu,\sigma^2

· Sample:

\bar{x},s^2

· Relative frequency = category count /

n

Mean and median

Arithmetic mean

\bar{x}=\frac1n\sum_{i=1}^n x_i\qquad \mu=\frac1N\sum_{i=1}^N x_i

Sorted data, odd n; j =

(n+1)/2

\textcolor{#8bded6}{\tilde{x}}=x_j

Sorted data, even n; j =

n/2

\textcolor{#8bded6}{\tilde{x}}=\frac{x_j+x_{j+1}}2

Balance point and middle position

Meaning

The mean is the arithmetic balance point. After sorting, $x_j$ is the $j$ th observation and the median $\tilde{x}$ divides the data into lower and upper halves.

When to use it

Use the mean for an arithmetic average. Use the median for a resistant center when skew or extreme observations matter.

Skew follows the long tail. Mean–median ordering is a typical pattern, not a universal rule.

Changes to a dataset

Add a new observation y

\bar{x}_{\mathrm{new}}=\frac{\textcolor{#f4dc88}{n\bar{x}}+\textcolor{#8bded6}{y}}{n+1}

Remove an observed value y, with

n>1

\bar{x}_{\mathrm{new}}=\frac{n\bar{x}-y}{n-1}

Replace an observed x by y

\bar{x}_{\mathrm{new}}=\bar{x}+\frac{y-x}{n}

Track the total and the count separately

Meaning

The original total is $n\bar{x}$ . Update that total and the number of observations, then divide to find the new mean.

When to use it

Use for adding, removing, or replacing data values. To update the median, re-sort and locate the new middle position or positions.

The median depends on order. The original mean and sample size alone do not determine the new median.

Quartiles and the five-number summary

Ordered observations

x_1\le x_2\le\cdots\le x_n

Course quartile positions

\textcolor{#8bded6}{n_1}=\frac{n+1}{4}\qquad\textcolor{#ee9daf}{n_3}=\frac{3(n+1)}4

Interpolation at position j + d

Q=x_j+\textcolor{#f4dc88}{d}(x_{j+1}-x_j)

A fractional position interpolates between neighbors

Meaning

The five-number summary is minimum, $Q_1$ , median $\tilde{x}$ , $Q_3$ , maximum. The quartiles mark the lower and upper quarters.

When to use it

For a noninteger position, move fraction $d$ of the way from observation $j$ to observation $j+1$ . At an integer position, use that observation.

This is the course convention (

n\ge3

). Software can use different quartile definitions.

IQR and outlier boxplots

Two measures of spread

\textcolor{#f4dc88}{\mathrm{IQR}}=\textcolor{#ee9daf}{Q_3}-\textcolor{#8bded6}{Q_1}\qquad \mathrm{range}=\max-\min

Lower fence

L=\textcolor{#8bded6}{Q_1}-1.5\,\mathrm{IQR}

Upper fence

U=\textcolor{#ee9daf}{Q_3}+1.5\,\mathrm{IQR}

Box, whiskers, fences, and flagged points

Meaning

The box runs from $Q_1$ to $Q_3$ , with a line at the median. Whiskers end at the most extreme observed values inside the fences.

When to use it

Use a boxplot to compare center, middle-half spread, asymmetry, and possible outliers across quantitative distributions.

Fences are cutoffs. Whiskers are observed values. Flagged observations need investigation.

Population and sample variance

Complete population of N values

\sigma^2=\frac{\sum_{i=1}^N(x_i-\mu)^2}{N}=\frac{\sum_{i=1}^Nx_i^2}{N}-\mu^2

Sample of

n>1

values

\textcolor{#f4dc88}{s^2}=\frac{\sum_{i=1}^n\textcolor{#8bded6}{(x_i-\bar{x})^2}}{n-1}

Computational form

s^2=\frac{\sum_{i=1}^nx_i^2-n\bar{x}^{\,2}}{n-1}

Squaring makes every deviation contribute positively

Meaning

Variance averages squared deviations from the relevant mean. SD is the square root: $\sigma=\sqrt{\sigma^2}$ and $s=\sqrt{s^2}$ .

When to use it

Choose the denominator from the task: a complete population, an estimated population variance, or a probability model.

Variance has squared units. SD has original units. An empirical distribution uses

1/n

weights.

Shifts, scales, and standardized values

Transform every observation

y_i=\textcolor{#8bded6}{a}x_i+\textcolor{#ee9daf}{b}

Center and spread

\bar{y}=a\bar{x}+b\qquad s_y^2=a^2s_x^2

Standardized distance: population / sample

\textcolor{#f4dc88}{z}=\frac{x-\mu}{\sigma}\qquad z=\frac{x-\bar{x}}s

Shift moves the center; scale changes the spread

Meaning

A shift changes the center. Scaling by $a$ multiplies SD by $|a|$ and variance by $a^2$ . A $z$ -score measures distance from the mean in SD units.

When to use it

Use these rules for unit conversions, adding a constant, or comparing relative position. Standardization requires positive SD.

For an approximately normal shape: about 68%, 95%, 99.7% lie within 1, 2, 3 SDs.

Covariance

Sample paired observations

\textcolor{#f4dc88}{s_{xy}}=\frac{\sum_{i=1}^n\textcolor{#8bded6}{(x_i-\bar{x})}\textcolor{#ee9daf}{(y_i-\bar{y})}}{n-1}

Complete paired population

\sigma_{xy}=\frac1N\sum_{i=1}^N(x_i-\mu_x)(y_i-\mu_y)

Computational form

s_{xy}=\frac{\sum_{i=1}^nx_iy_i-n\bar{x}\bar{y}}{n-1}

Centered coordinates reveal each contribution

Meaning

Covariance averages products of paired deviations. Same-direction deviations contribute positively. Units are $x$ -units times $y$ -units.

When to use it

Use it to quantify joint direction, compute correlation, or build the least-squares slope. Keep each observed pair intact.

A scatterplot reveals shape and unusual points that a single number can hide.

Correlation

Standardized covariance

\textcolor{#f4dc88}{r}=\frac{s_{xy}}{\textcolor{#8bded6}{s_x}\textcolor{#ee9daf}{s_y}}

Population counterpart

\rho=\frac{\sigma_{xy}}{\sigma_x\sigma_y}

Both SDs must be positive

-1\le r,\rho\le1\qquad |r|=1:\ \text{exact linear association}

The middle pattern is nonlinear despite zero correlation

Meaning

Correlation gives the direction and strength of linear association. It is unitless. A larger $|r|$ means a closer linear relationship.

When to use it

Use $r$ to compare linear association across scales. Inspect the scatterplot for curvature and influential observations first.

With positive SDs:

\operatorname{sign}(r)=\operatorname{sign}(s_{xy})=

sign of the centered cross-product sum.

r=0

does not imply independence. Correlation alone does not establish causation.

Least-squares regression

Predict y using x

\textcolor{#f4dc88}{\hat{y}}=\textcolor{#8bded6}{m}x+b

Slope

\textcolor{#8bded6}{m}=\frac{s_{xy}}{s_x^2}=r\frac{s_y}{s_x}

Intercept (prediction at

x=0

) and residual

b=\bar{y}-m\bar{x}\qquad\textcolor{#ee9daf}{e_i}=y_i-\hat{y}_i

Residual = observed minus fitted

Meaning

$x$ is the explanatory variable and $y$ is the response. The slope gives predicted $y$ change per $x$ unit. A positive residual means observed $y$ exceeds predicted $y$ .

When to use it

Use the line for linear prediction with variation in $x$ . Least squares minimizes $\sum_i e_i^2$ and the fitted line passes through $(\bar{x},\bar{y})$ .

Slope requires

s_x>0

. The

r

form also needs

s_y>0

. Extrapolation needs justification.

Events and set operations

Both events occur

\textcolor{#8bded6}{A}\cap\textcolor{#ee9daf}{B}

At least one occurs

A\cup B

A occurs without B

A\cap B^c

The highlighted region follows the current formula

Meaning

$S$ is the sample space. An event is a subset of $S$ . $A^c$ contains outcomes outside $A$ . Disjoint events have $A\cap B=\varnothing$ .

When to use it

“And” suggests an intersection. Inclusive “or” suggests a union. “Neither” means $A^c\cap B^c$ . “Exactly one” excludes the overlap.

(A\cup B)^c=A^c\cap B^c\qquad(A\cap B)^c=A^c\cup B^c

Probability rules

Basic constraints

P(S)=1\qquad0\le P(A)\le1

Complement

P(A^c)=1-P(A)

Union

P(A\cup B)=\textcolor{#8bded6}{P(A)}+\textcolor{#ee9daf}{P(B)}-\textcolor{#f4dc88}{P(A\cap B)}

Subtract the overlap counted twice

Meaning

The union formula subtracts the overlap counted twice. For disjoint events, the overlap is empty and probabilities add.

When to use it

“At least one” is often easiest as one minus “none.” For finite equally likely outcomes only, $P(A)=\#A/\#S$ .

P(A\cap B^c)=P(A)-P(A\cap B)

Counting by the sampling mechanism

Ordered, with replacement

n^k\qquad\text{each position has }n\text{ choices}

Ordered, without replacement: available choices decrease

(n)_k=\frac{n!}{(n-k)!}

Unordered, without replacement: remove

k!

reorderings

\binom{n}{k}=\frac{n!}{k!(n-k)!}

Replacement and order determine the count

Meaning

The product rule multiplies the number of choices at successive stages. Add counts when alternatives are disjoint.

When to use it

“Arrange” or “sequence” makes order matter. “Choose a group” usually makes order irrelevant. Replacement keeps objects available.

0!=1\qquad\binom{n}{k}=\binom{n}{n-k}

· Numerator and denominator must count the same objects.

Conditional probability and multiplication

Restriction to a reference event

P(A\mid\textcolor{#ee9daf}{B})=\frac{\textcolor{#f4dc88}{P(A\cap B)}}{\textcolor{#ee9daf}{P(B)}}

Multiplication rule

P(A\cap B)=P(B)P(A\mid B)

Several stages

P(A\cap B\cap C)=P(A)P(B\mid A)P(C\mid A\cap B)

The given event B is the new reference group

Meaning

Conditioning keeps outcomes in $B$ and rescales their total probability to one. The denominator is the group named after “given.”

When to use it

Use conditional probabilities when earlier outcomes change later chances, or when the question restricts the reference group.

Conditioning events need positive probability. The multiplication rule does not require independence.

Independence

Definition for two events

P(A\cap B)=\textcolor{#8bded6}{P(A)}\textcolor{#ee9daf}{P(B)}

Equivalent when

P(B)>0

P(A\mid B)=P(A)

Mutual independence: every subcollection

P(A_{i_1}\cap\cdots\cap A_{i_k})=\prod_{j=1}^kP(A_{i_j})

An independent area model: overlap equals a product

Meaning

Independence means learning one event occurred does not change the probability of the other. Disjointness means both cannot occur.

When to use it

Use a product of marginal probabilities only when independence follows from the model or has been established.

Positive-probability disjoint events are dependent. Pairwise independence alone is insufficient for 3 or more events.

Law of total probability

A partition of the sample space

\bigcup_{i=1}^{m}B_i=S\qquad B_i\cap B_j=\varnothing\ (i\ne j)

Probability over all routes

\textcolor{#f4dc88}{P(A)}=\sum_{i=1}^{m}\textcolor{#8bded6}{P(A\mid B_i)}\textcolor{#ee9daf}{P(B_i)}

Every group contributes one route to A

Meaning

Each term is the probability of one route into $A$ : first belong to group $B_i$ , then satisfy $A$ within that group.

When to use it

Use when a population or experiment splits into distinct groups, mechanisms, or starting states with different conditional chances.

Use exhaustive, nonoverlapping groups. Omit zero-probability groups when writing conditionals.

Bayes’ rule

Reverse the conditioning direction

P(B_j\mid A)=\frac{\textcolor{#8bded6}{P(A\mid B_j)}\textcolor{#ee9daf}{P(B_j)}}{\textcolor{#f4dc88}{P(A)}}

Expand the evidence probability

P(B_j\mid A)=\frac{P(A\mid B_j)P(B_j)}{\sum_{i=1}^{m}P(A\mid B_i)P(B_i)}

Which starting group produced the evidence?

Meaning

Prior $P(B_j)$ : group probability before evidence. Likelihood $P(A\mid B_j)$ : evidence probability within the group. Posterior: group probability after evidence.

When to use it

Use when evidence is observed and the question asks which group or source produced it. Work from the known conditioning direction.

P(A)>0

. The denominator includes every route that can produce

A

.

Discrete random variables and PMFs

Probability mass function

\textcolor{#f4dc88}{f(x)}=P(X=x)\ge0\qquad\sum_{\text{all }x}f(x)=1

Event probability: sum over support values in A

P(X\in A)=\sum_{x\in A}f(x)

Cumulative distribution function

\textcolor{#8bded6}{F(t)}=P(X\le t)=\sum_{x\le t}f(x)

Add only the masses in the requested event

Meaning

The support lists possible values with positive mass. Multiple experimental outcomes can contribute to the same value of $X$ .

When to use it

Define $X$ in words, list its support, and translate the question into a set of values. Then sum the corresponding masses.

Capital

X

is the random variable. Lowercase

x

is a possible value.

Expectation

Weighted mean

\textcolor{#f4dc88}{\mu_X}=E(X)=E[X]=\sum_{\text{all }x}\textcolor{#8bded6}{x}f(x)

A function of X

E[g(X)]=\sum_{\text{all }x}g(x)f(x)

Second moment

E(X^2)=\sum_{\text{all }x}x^2f(x)

Expectation is the probability-weighted balance point

Meaning

Expectation is a probability-weighted average. $E[g(X)]$ averages the transformed values using the original probabilities.

When to use it

Use for a long-run average, expected count, or average value after a transformation. The mean need not be a possible outcome.

E[g(X)]

generally differs from

g(E(X))

. Infinite-support formulas need the relevant moments to exist.

Variance of a random variable

Definition

\textcolor{#f4dc88}{\sigma_X^2}=V(X)=\operatorname{Var}(X)=E\!\left[(X-\mu_X)^2\right]

PMF form

V(X)=\sum_{\text{all }x}\textcolor{#8bded6}{(x-\mu_X)^2}f(x)

Computational form

V(X)=E(X^2)-[E(X)]^2

Weight squared distances by probability

Meaning

Variance is the weighted mean squared distance from the mean. It is nonnegative. $\sigma_X=\sqrt{V(X)}$ returns to the original units.

When to use it

Use the definition to interpret spread. Use the second-moment identity when $E(X)$ and $E(X^2)$ are easier to compute.

Finite second moment assumed. Model probability weights require no

n-1

correction.

Linear combinations: means and variances

Linearity of expectation

E(aX+bY+c)=aE(X)+bE(Y)+c

Variance with dependence retained

\begin{aligned}V(aX+bY+c)&=a^2V(X)+b^2V(Y)\\&\quad+\textcolor{#ee9daf}{2ab\operatorname{Cov}(X,Y)}\end{aligned}

Dependence contributes the cross term

Meaning

Expectations add without independence. Variance also contains a covariance term. A constant shift contributes no variance.

When to use it

Use for totals, differences, weighted combinations, and changes of units. Retain the covariance term unless it is zero.

For finite second moments:

V(aX+c)=a^2V(X)

and

\sigma_{aX+c}=|a|\sigma_X

.

Covariance, sums, and averages

Model covariance

\operatorname{Cov}(X,Y)=E(XY)-E(X)E(Y)

Variance of a sum: every pair

i<j

V\!\left(\sum_i X_i\right)=\sum_iV(X_i)+\textcolor{#ee9daf}{2\sum_{i<j}\operatorname{Cov}(X_i,X_j)}

Independent, identically distributed values

E(\overline{X})=\mu\qquad\textcolor{#f4dc88}{V(\overline{X})}=\frac{\sigma^2}{n}

For iid values, averaging reduces variance by n

Meaning

Independent variables have $E(XY)=E(X)E(Y)$ and zero covariance. Zero covariance alone does not establish independence.

When to use it

Use the sum formula to track dependence. For an average of $n$ independent equal-variance measurements, $\sigma_{\overline{X}}=\sigma/\sqrt{n}$ .

E\!\left(\sum_i a_iX_i\right)=\sum_i a_iE(X_i)

· No independence is needed for this mean formula.

Bernoulli variables and indicators

One binary outcome

I=\begin{cases}1&\text{on }A\\0&\text{on }A^c\end{cases}

Success and failure probabilities

P(I=1)=\textcolor{#8bded6}{p}\qquad P(I=0)=\textcolor{#ee9daf}{q}=1-p

Moments

E(I)=p\qquad V(I)=pq

The bar heights are schematic: p and q sum to one

Meaning

An indicator records whether an event occurs. “Success” is simply the outcome chosen for counting, and $0\le p\le1$ .

When to use it

Use Bernoulli for one yes/no outcome. Write a count as $X=\sum_i I_i$ to compute $E(X)=\sum_iP(A_i)$ , even with dependence.

A sum of indicators requires extra assumptions to have a binomial distribution.

Independent indicators with different probabilities

One success set J;

q_i=1-p_i

w(J)=\prod_{i\in J}\textcolor{#8bded6}{p_i}\prod_{\substack{1\le i\le n\\i\notin J}}\textcolor{#ee9daf}{q_i}

Sum over sets

J\subseteq\{1,\ldots,n\}

with

|J|=k

P(X=k)=\sum_{\substack{J\subseteq\{1,\ldots,n\}\\|J|=k}}w(J)

Moments

E(X)=\sum_i p_i\qquad V(X)=\sum_i p_iq_i

Each pattern has its own product weight

Meaning

Let independent $I_i$ be 1 with probability $p_i$ and 0 otherwise, with $0\le p_i\le1$ . $X=\sum_i I_i$ counts the successes.

When to use it

Use for independent trials with different success chances. Each $J$ names a success pattern. Add the weights for all patterns with $k$ successes.

Ordinary binomial requires equal

p_i

. Their average preserves the mean but generally gives wrong exact-count probabilities.

Binomial distribution

Count of successes;

q=1-p

X\sim\operatorname{Binomial}(n,p)\qquad k=0,1,\ldots,n

Probability mass function

\textcolor{#f4dc88}{f(k)}=P(X=k)=\binom{n}{k}\textcolor{#8bded6}{p^k}\textcolor{#ee9daf}{q^{n-k}}

Moments

\mu_X=E(X)=np\qquad\sigma_X^2=V(X)=npq

A fixed number of independent, common-p trials

Meaning

$\binom{n}{k}$ chooses the success positions. The powers give the probability of one arrangement with $k$ successes and $n-k$ failures.

When to use it

Use when there are a fixed $n$ binary trials, mutual independence, and the same success probability $p$ at every trial.

\binom{n}{k}=\dfrac{n!}{k!(n-k)!}

· “Counts successes” alone does not establish the model.

Translating count questions

Exactly k

P(X=k)=\textcolor{#f4dc88}{f(k)}

At most k

P(X\le k)=\sum_{j\le k}f(j)

At least k

P(X\ge k)=1-P(X\le k-1)

Exactly k → at most k → at least k

Meaning

For integer counts: “fewer than $k$ ” means $X\le k-1$ . “More than $k$ ” means $X\ge k+1$ . “At least one” is the complement of zero.

When to use it

Use the chosen model’s PMF inside the sum. Limit the summation to its valid support, then use a complement if that is simpler.

P(a\le X\le b)=F(b)-F(a-1)

for integers

a\le b

.

Hypergeometric distribution

Sample without replacement

X\sim\operatorname{Hyp.Geo.}(N,K,n)

Probability mass function

P(X=k)=\frac{\textcolor{#8bded6}{\binom{K}{k}}\textcolor{#ee9daf}{\binom{N-K}{n-k}}}{\binom{N}{n}}

Moments;

p=K/N

and

q=1-p

\mu_X=np\qquad\sigma_X^2=npq\,\textcolor{#f4dc88}{\frac{N-n}{N-1}}

Choose successes and failures from their own groups

Meaning

$N$ objects contain $K$ successes. A uniformly random sample of $n$ distinct objects gives $X$ successes. The draws are generally dependent.

When to use it

Use for a fixed-size sample without replacement from a known finite population. The numerator chooses both successes and failures.

\max(0,n-(N-K))\le k\le\min(n,K)

· Variance formula assumes

N>1

.

Poisson distribution

Count in a specified exposure;

\lambda>0

X\sim\operatorname{Poisson}(\textcolor{#f4dc88}{\lambda})\qquad k=0,1,2,\ldots

Probability mass function

f(k)=P(X=k)=e^{-\lambda}\frac{\lambda^k}{k!}

Moments and exposure

\mu_X=E(X)=\sigma_X^2=V(X)=\lambda\qquad\lambda=rt

Rate × exposure gives the expected count

Meaning

$\lambda$ is the expected count for the stated exposure. Under a homogeneous Poisson process, $r$ is the constant rate and $t$ is the exposure length.

When to use it

Use for event counts under a model with independent disjoint increments and rare single events in very small intervals.

Match rate and exposure units. An average rate by itself does not justify a Poisson model.

Choosing a discrete model

Bernoulli: one binary trial

p\quad(q=1-p)

Binomial: fixed n, independent, common p

n,p

Hypergeometric: n distinct draws without replacement

N,K,n

Poisson: count over a stated exposure

\lambda=\text{expected count}

Start with the mechanism, then select the distribution

Meaning

The sampling mechanism determines the distribution. The desired event then determines a PMF term, sum, or complement.

When to use it

Identify what $X$ counts, its support, the parameters, and which assumptions the wording actually supplies.

Approximations: hypergeometric

\approx

binomial for a small sampling fraction; binomial

\approx

Poisson for large

n

, small

p

.

A reliable setup and answer check

1

Identify the object

A statistic, an event, or a random variable

2

Translate the wording

The reference group, support, and event endpoints

3

Justify the formula

The sampling mechanism and required assumptions

4

Interpret and check

Probability bounds, units, signs, and plausibility

A setup that explains why the formula applies