Problem-solving recipes

A step-by-step procedure for every problem type this course teaches — what to do first, what to do next, and how to tell which case you're in.

What are you being asked to do?

Start with what you're actually being asked to produce.

A sum written in sigma notation, need its value — the readiness recipe.

A joint probability table, need to check it, or extract something from it — verify or complete it first, then whichever of marginal, conditional, or conditional-expectation matches what's actually asked.

Two random variables from a joint table, need to know if they're independent — the independence-checking recipe.

A joint distribution, need covariance or correlation — covariance first; correlation extends it with one more division step.

A random variable under a linear transformation $Y=aX+b$ , need its mean, or its mean and variance — the mean recipe alone, or the extended mean-and-variance recipe once variance is also asked for.

A sum of independent random variables, need its mean, variance, or SD — the sum recipe (not the same as the linear-transformation recipes above, which are for scaling one variable, not adding several).

Why an early run of luck doesn't get "corrected" as more trials come in — the diluted-streak recipe.

A sample's standard error, need it computed or need to know how it changes as $n$ changes — the standard-error recipe. From a finite population sampled without replacement, apply the finite-population correction on top of it instead.

A sample proportion, need to approximate a probability about it — the sampling-distribution-of-a-proportion recipe.

How many people needed before two share a birthday, likely — the birthday-collision recipe.

Readiness

Evaluate a sum written in sigma notation

Use when: you see $\sum$ with a lower index, an upper index, and a formula for the general term, and need a single number

Goal: the exact number the sum evaluates to

Sigma notation and finite sums

Joint distributions

Verify or complete a joint probability table

Use when: you're given a table of joint probabilities, possibly with one entry unknown, and asked to confirm it's valid or find the missing value

Goal: either a confirmed valid joint pmf, or the missing cell's value that makes the table valid

Joint probability mass functions

Find a marginal distribution from a joint table

Use when: you have a joint probability table and need one variable's own distribution by itself

Goal: the complete list of probabilities for one variable, each obtained by summing across the joint table

Marginal distributions

Find a conditional distribution from a joint table

Use when: you're told one variable's value is given, and asked for the OTHER variable's whole distribution under that condition

Goal: a complete, valid probability distribution for the other variable, conditioned on the given value

Conditional distributions

Needs a marginal total from the recipe above as part of the division step.

Compute a conditional expectation

Use when: you need $E(Y\mid X=x)$ and have (or can build) the joint distribution of $X$ and $Y$

Goal: a single number: $Y$ 's average value, restricted to the world where $X=x$

Conditional expectation

Needs a conditional distribution from the recipe above first.

Check whether two random variables are independent

Use when: you have a full joint table for $X$ and $Y$ and need a definitive verdict, yes or no, on whether $p(x,y)=p_X(x)p_Y(y)$ holds at every cell

Goal: either a confirmed yes, checked at every cell, or a specific cell that disproves it

Independence of random variables

Compute covariance from a joint distribution

Use when: you have a full joint table (or the raw pieces $E(XY)$ , $E(X)$ , $E(Y)$ ) and need $\operatorname{Cov}(X,Y)$

Goal: a single signed number measuring how $X$ and $Y$ move together, ready to read by its sign, or to rescale into correlation later

Covariance · extended later in this course

Compute correlation from a joint distribution

Use when: you have a full joint table (or covariance plus both standard deviations) and need $\rho_{X,Y}$

Goal: a single unitless number between $-1$ and $1$ measuring the strength and direction of the linear relationship between $X$ and $Y$

Correlation

Extends the covariance recipe above with one more division step and a bound check.

Transformations of random variables

Find the mean after a linear transformation

Use when: you have $Y=aX+b$ and are asked either for $E(Y)$ , or for a missing constant $a$ or $b$ that makes $E(Y)$ hit a target value

Goal: either $E(Y)$ itself, or the specific constant that produces a target mean

Mean under a linear transformation · extended later in this course

Find the mean and variance after a linear transformation

Use when: you have $Y=aX+b$ and need $E(Y)$ , $\operatorname{Var}(Y)$ , or a missing constant, from what's known about $X$

Goal: the complete pair $E(Y)$ and $\operatorname{Var}(Y)$ (or a missing constant), using the linear-transformation rules for mean and variance together

Variance under a linear transformation

Extends the mean-only recipe above once variance is also asked for.

Find the mean, variance, and SD of a sum of independent random variables

Use when: you're asked for $E(Y)$ , $\operatorname{Var}(Y)$ , or the standard deviation of a sum $Y=X_1+X_2+\cdots$ of random variables

Goal: the sum's own mean, variance, and, if needed, standard deviation, built directly from each piece's own mean and variance

Sums of independent random variables

For adding several independent variables together, not for scaling one variable, a different problem from the linear-transformation recipes above.

Limit theorems and sampling

Quantify how an early streak gets diluted, not corrected

Use when: you're given a fixed imbalance (a streak or run) in the first few trials of a repeated random process, plus more trials assumed to split at the expected rate. You're asked how the running proportion or count behaves after that

Goal: the running proportion after the additional trials, plus confirmation that the streak's raw count imbalance is still exactly present, unchanged

Why probability doesn't "balance out" immediately

Compute a standard error, and how it changes with n

Use when: you know a population standard deviation $\sigma$ and a sample size $n$ , and need $SE$ itself, or need to know how $SE$ changes after $n$ changes

Goal: the standard error itself, or a correctly scaled new $SE$ after a change in $n$

Standard error

Apply the finite population correction to a standard error

Use when: you're computing a standard error for sampling WITHOUT replacement from a finite population of known size $N$

Goal: the corrected standard error, or a justified decision that the correction is small enough to skip

The finite population correction

Builds on the standard-error recipe above, adding a correction factor for sampling without replacement from a finite population.

Approximate a probability for a sample proportion

Use when: you know a population proportion $p$ and a sample size $n$ , and need either the shape of $\hat p$ 's sampling distribution or a probability involving $\hat p$

Goal: the normal approximation's mean and standard error for $\hat p$ , and, if needed, a probability found by standardizing and reading $\Phi$

The sampling distribution of a proportion

Probability in the wild

Compute a birthday-collision probability

Use when: you're asked for the probability that at least two people (or items) out of $n$ share a birthday, or land in the same one of $D$ equally likely categories

Goal: the exact collision probability, built by first computing the NO-collision product and then subtracting it from 1

The birthday problem