Pretty straightforward if you've reviewed the basics.
Start by defining what 'errors' or 'residuals' mean in each model, then contrast their properties: in linear regression, residuals are the differences between observed and predicted continuous values, assumed to be normally distributed with constant variance; in logistic regression, residuals are typically defined on the log-odds or probability scale, and the concept of a single residual is less straightforward due to the binary outcome. Emphasize that the key difference lies in the nature of the response variable and the assumptions about the error distribution.
Pro tip: Mention that in logistic regression, deviance residuals are often used instead of raw residuals, and that the focus is on the likelihood rather than squared error. This shows depth beyond textbook definitions.
Explain that residuals are the differences between observed and predicted continuous outcomes, and that they are assumed to be independent and normally distributed with constant variance.
Clarify that the response is binary, so residuals are not as intuitive; common types include response residuals (observed minus predicted probability) and deviance residuals, which measure the contribution to the likelihood.
Highlight that linear regression minimizes squared error and assumes homoscedasticity, while logistic regression maximizes likelihood and does not assume constant variance of residuals; instead, it models the log-odds.
Note that in linear regression, residual plots are used to check assumptions, whereas in logistic regression, deviance residuals or Pearson residuals are used, and patterns can indicate lack of fit.
Conclude that the correct description is that linear regression residuals are on the outcome scale and assumed normal, while logistic regression residuals are on the probability or log-odds scale and are not normally distributed.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
Model the process as a Markov chain with states representing the current streak of consecutive heads (0, 1, 2, 3). Set up equations for the expected number of tosses from each state and solve for the expected value starting from 0 heads.
Pro tip: After solving, verify your answer by simulation or by checking that the expected value is consistent with the probability of getting three heads in a row in a given number of tosses. Also, be prepared to discuss the memoryless property and how it simplifies the equations.
Identify the states based on the current streak of consecutive heads: 0, 1, 2, and 3 (absorbing state).
Let E_i be the expected number of additional tosses to reach 3 heads in a row from state i. Write equations: E_3 = 0, E_2 = 1 + 0.5 E_3 + 0.5 E_0, E_1 = 1 + 0.5 E_2 + 0.5 E_0, E_0 = 1 + 0.5 E_1 + 0.5 E_0.
Solve the system of linear equations to find E_0. From E_0 equation: E_0 = 2 + E_1. Substitute into E_1 and E_2 to get E_1 = 6, E_2 = 10, and finally E_0 = 14.
State that the expected number of tosses is 14. Optionally, verify by simulation or by using the formula for expected waiting time for a pattern.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.