Some Problem-Solving Strategies

Drawing plots - 1 variable

Determine the type of plot
- 1 variable: sketch a pdf plot, height represents density
- 2 variables: sketch a joint pdf/scatter plot, shading represents density
Determine possible values of RVs and label the variable axis
Consider a few possible values and sketch a few points
Think about where density will be higher/lower and sketch the plot
Know specific shapes corresponding to named distributions (e.g., Exponetial)
Know the difference between discrete and continuous RVs.
If continuous, it often helps to consider a discrete analog first. But make sure you don’t stop at just the discrete analog.

Drawing plots - 2 variables

Determine possible values of RVs and label the variable axes
Even if values of one RV conditionally depend on the other, identify overall possible values of each RV
If values of one RV conditionally depend on the other, draw the “boundaries” of possible pairs
Consider a few possible values and sketch a few points
Think about where density will be higher/lower and shade the plot
If set up provides conditional distribution of one variable given the other (say $Y$ given $X$ ), sketch the plot in two stages
- First, plot the $X$ values according to marginal of $X$ (to determine the X “stacks”)
- Then, for each $x$ stack distribute the $Y$ values along each vertical slice according to the conditional distribution of $Y$ given $X = x$
- When doing the previous step, consider a few specific numerical values for $x$ ( $x = 1$ , etc)
For conditional or marginal plots, start by sketching a good joint pdf
For conditional of $X$ given $Y = y$ : slice the joint pdf by fixing $y$ values
- Identify the possible values of $X$ along the $Y = y$ slice
- Conditional distribution of $X$ given $Y = y$ is a distribution on $x$ values only. If $y$ shows up, it’s treated like a constant. So this becomes a 1 variable situation
For marginal of $Y$ : for each $y$ collapse/stack the joint pdf over the $X$ values.
- Remember: marginal distribution of $Y$ is a distribution on $y$ values only. Even if the values of $Y$ conditionally depend on $X$ , the marginal distribution of $Y$ should only include the overall possible values of $Y$ , and no $x$ s.
- What values of $y$ correspond to longer intervals of possible $x$ values? Stacking over a longer interval (i.e., more stacks) will yield higher density at $y$
- For what values of $y$ is the density along the slice higher? Stacking higher stacks will yield higher density at $y$

Expected values

Find “expected number of” try indicators; often an $n p$ formula works
Linearity of expected value $E (X) + E (Y)$ always works regardless of it $X$ and $Y$ are related
Use LTE $E (Y) = E (E (Y | X))$ , especially if the set up tells how you $Y$ depends on $X$
If it seems like there are different cases, consider each case separately and use LTE
If a problem involves the expected number of stages or rounds or iterations, try conditioning on the first stage/round/iteration and set up an equation to solve for the expected value
Know expected values and variance/SD formulas for all the named distributions
$E (X^{2}) = V a r (X) + (E (X^{2}))$
Know what the answer should look like.
- $E (Y)$ is a number.
- $E (Y | X = 2)$ is a number.
- $E (Y | X)$ is a random variable, and a function of $X$ .
LOTUS for expected values of functions: $E (g (X)) = \int_{\infty}^{\infty} g (x) f_{X} (x) d x$ . (Be sure to replace bounds with possible values of $X$ and don’t mess with the pdf $f_{X} (x)$ .)
If you can’t think of anything else, try the definition of expected value: you’ll need to list possible values and find the pmf/pdf.
- But this should generally not be your first strategy, unless the pmf/pdf is provided

Covariance

If a problem involves $E (X Y)$ think $C o v (X, Y)$ and vice verse
Use LTE and TOWIK to find $E (X Y) = E (X E (Y | X))$ especially if the set up tells how you $Y$ depends on $X$
If $X$ and $Y$ are independent then the covariance is 0 (but the converse is not true)

Conditioning

If a problem involves two variables/events, use two-way tables of counts
For a joint pdf, fix one of the variables to find conditional distributions.
- Fix a value of $x$ to find the conditional distribution of $Y$ given $X = x$ .
- Plug in specific numbers for $x$ first, then find the general pattern
If a problem seems to involve multiple cases, consider each case separately and use the law of total probability
See also the LTE strategies in the expected value section

General strategies

Always identify possible values of RVs
Know what it means for an RV to have a particular named distribution. Know the formulas, but also know what would simulated data look like?
Draw pictures, even if the problem doesn’t ask you to. You can often find probabilities via geometry
Identify if a problem involves discrete or continuous RVs, and know the difference, e.g., pmf and sums for discrete, pdf and integrals for continuous
- If continuous, it often helps to consider a discrete analog first. But make sure you don’t stop at just the discrete analog.
If you can’t think of anything else, try listing possible outcomes. Considering a few possible outcomes often helps make the problem more concrete
- But trying to list all possible outcomes should not be your first strategy
Look for independence. If the problem says “independent” take advantage. Given a joint pdf/pmf, see if it factors - but don’t forget the possibel values.
Look for named distributions. If you can recognize the “meat” of distributions like Exponential, Poisson, etc, you can use the shortcut formulas for probabilities, EV, variance