Answer basic probability and statistics questions
Probability and Statistics Questions
This is a set of independent probability and statistics questions, typical of a data science or machine learning screening.
1. Ordering Poisson Distributions by Rate Parameter
Three Poisson-distributed random variables, XA, XB, and XC, have their probability mass functions (PMFs) plotted on the same graph (support on non-negative integers). The distributions are described as follows:
- Distribution A: Most probability mass is concentrated on values 0, 1, and 2. The mode is at 1. Probability at 0 is high, and probabilities drop off quickly after 3.
- Distribution B: The mode is at 3. The distribution is more spread out than A, with noticeable probability up to around 6 or 7, but very little after that.
- Distribution C: The mode is at 5. The distribution is the most spread out of the three, with noticeable probability from around 2 up to 10 or more.
All three are Poisson distributions with parameters λA, λB, and λC, respectively.
Question 1: Based on these qualitative descriptions, order the three rate parameters (λA, λB, λC) from smallest to largest. Choose the correct ordering and briefly justify your choice.
2. Identifying Type of Bias in a Study
A startup incubator wants to understand what makes startups successful. They collect data only from companies that have already raised Series C or later funding and are still operating. They analyze features like team size, prior founder experience, average founder age, and industry, and then publish a report claiming these are the characteristics that make startups successful.
Question 2: What is the primary type of bias in this study? Choose the best option and briefly explain.
- (a) Survivor (survivorship) bias
- (b) Sampling bias (non-representative sample)
- (c) Recall bias
- (d) No bias; the study design is appropriate
(You may mention more than one kind of bias if relevant, but identify the primary/statistically standard name.)
3. True/False Questions on Covariance and Correlation
For each of the following statements about covariance and correlation between two real-valued random variables X and Y, answer True or False and provide a brief justification.
- Statement A: If Cov(X,Y) = 0, then X and Y are independent.
- Statement B: The Pearson correlation coefficient ρXY is always between -1 and 1, inclusive.
- Statement C: The Pearson correlation coefficient between X and Y is given by ρXY = Cov(X,Y) / (σX * σY), where σX and σY are the standard deviations of X and Y, respectively.
- Statement D: If we rescale X by a positive constant a > 0, i.e., define X' = aX, then the correlation between X' and Y is the same as the correlation between X and Y.
- Statement E: A very high correlation between X and Y implies that X causes changes in Y.
State True/False for each and justify in one or two sentences.
AI interview prep
Core dilemma
Assessing fundamental understanding of statistical distributions and common biases versus rote memorization.
3 tradeoffs and 3 stress tests locked
Interview prep
Unlock the interviewer X-Ray rubric, tradeoff matrix, stress tests, clarifying questions, and key concepts.
Answer basic probability and statistics questions
Probability and Statistics Questions
This is a set of independent probability and statistics questions, typical of a data science or machine learning screening.
1. Ordering Poisson Distributions by Rate Parameter
Three Poisson-distributed random variables, XA, XB, and XC, have their probability mass functions (PMFs) plotted on the same graph (support on non-negative integers). The distributions are described as follows:
- Distribution A: Most probability mass is concentrated on values 0, 1, and 2. The mode is at 1. Probability at 0 is high, and probabilities drop off quickly after 3.
- Distribution B: The mode is at 3. The distribution is more spread out than A, with noticeable probability up to around 6 or 7, but very little after that.
- Distribution C: The mode is at 5. The distribution is the most spread out of the three, with noticeable probability from around 2 up to 10 or more.
All three are Poisson distributions with parameters λA, λB, and λC, respectively.
Question 1: Based on these qualitative descriptions, order the three rate parameters (λA, λB, λC) from smallest to largest. Choose the correct ordering and briefly justify your choice.
2. Identifying Type of Bias in a Study
A startup incubator wants to understand what makes startups successful. They collect data only from companies that have already raised Series C or later funding and are still operating. They analyze features like team size, prior founder experience, average founder age, and industry, and then publish a report claiming these are the characteristics that make startups successful.
Question 2: What is the primary type of bias in this study? Choose the best option and briefly explain.
- (a) Survivor (survivorship) bias
- (b) Sampling bias (non-representative sample)
- (c) Recall bias
- (d) No bias; the study design is appropriate
(You may mention more than one kind of bias if relevant, but identify the primary/statistically standard name.)
3. True/False Questions on Covariance and Correlation
For each of the following statements about covariance and correlation between two real-valued random variables X and Y, answer True or False and provide a brief justification.
- Statement A: If Cov(X,Y) = 0, then X and Y are independent.
- Statement B: The Pearson correlation coefficient ρXY is always between -1 and 1, inclusive.
- Statement C: The Pearson correlation coefficient between X and Y is given by ρXY = Cov(X,Y) / (σX * σY), where σX and σY are the standard deviations of X and Y, respectively.
- Statement D: If we rescale X by a positive constant a > 0, i.e., define X' = aX, then the correlation between X' and Y is the same as the correlation between X and Y.
- Statement E: A very high correlation between X and Y implies that X causes changes in Y.
State True/False for each and justify in one or two sentences.