Checkpoint: Distributions and Bayes

By Better Wise

Quiz

A column records the number of support tickets opened per hour at a help desk. Tickets arrive independently at a steady average rate, and the count can be any non-negative integer. Which distribution should you fit?
- Poisson, because it models counts of events in a fixed interval at a constant rate
- Binomial, because there is a fixed number of trials
- Gaussian, because counts are roughly symmetric around the mean
- Bernoulli, because each hour either has tickets or does not
`delivery_minutes` has mean 28 and median about 19.4. A colleague fits a Gaussian with mean 28 and standard deviation 16. What is the main problem with that choice?
- The Gaussian puts about 4% of probability below zero minutes and understates the long right tail
- The Gaussian median is always larger than its mean
- The Gaussian has two parameters while the exponential has one, so it is always worse
- The Gaussian cannot be standardized to a z-score
Which statements about the exponential model for a waiting time T with rate lambda are correct? Select all that apply.
- The mean waiting time is 1/lambda
- lambda is a duration in minutes, so a mean of 28 minutes means lambda = 28
- Memorylessness means P(T > s + t | T > s) = P(T > s)
- P(T > t) = e^(-lambda t)
- The median equals ln(2)/lambda, about 0.69 times the mean
Past orders give mu = 62 and sigma = 14 dollars. To find the share of orders above $80, you first convert the cutoff to a ____, computed as (80 - 62) / 14 = 1.29.
A payments rule flags transactions. The base rate of fraud is 1%, the rule catches 90% of fraud, and it falsely flags 5% of clean transactions. In a table of 10,000 transactions, 90 true positives and 495 false positives appear, so P(fraud | flag) = 90 / 585, which is about ____ percent.
A classifier reports 99% precision at 1% prevalence and flags a record. Using the odds form, prior odds are about 0.0101 and the likelihood ratio is 99. What is the posterior odds, and what probability does it correspond to?
- Posterior odds about 1.0, probability about 50%, which is far below the 99% precision figure
- Posterior odds about 1.0, so p = odds / (1 + odds) is about 50%, well below the quoted precision
- Posterior odds about 1.0, probability about 50%
- Posterior odds about 1.0, probability about 99%
Which of the following are correct practices when applying Bayes' rule to a screening question? Select all that apply.
- Build the 10,000-row natural-frequency table or multiply odds instead of relying on intuition
- Read a recall figure P(flagged | true) as if it were precision P(true | flagged)
- Ask for the base rate P(H) before looking at any test numbers
- Round a rare prior down to 0, since the event almost never happens
- At low prevalence, halving the false positive rate often moves the posterior more than raising sensitivity from 90% to 95%
The odds form of Bayes' rule says posterior odds equal prior odds multiplied by the ____, which is P(E | H) / P(E | not H).
A pressure sensor reports a leak with probability 0.98 when a leak exists and 0.02 when it does not. Historically, 0.5% of hourly readings are taken during a real leak. An alarm fires. Work through the full Bayes update and decide whether to shut the pipeline down. Explain your reasoning in a few sentences.
You are handed two columns from quickcart_orders.csv: order_value_usd (a measurement) and delivery_minutes (a waiting time). For each column, state which distribution you would fit, which summary statistics you would report, and one question you would answer with it. Justify your model choice from the shape and support of the column.