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This is a medium difficulty Machine Learning question, commonly asked during Take-home Project rounds at Morgan Stanley.

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This question is commonly asked for Data Scientist candidates at Morgan Stanley during technical interviews.

Answer basic probability and statistics questions | Morgan Stanley Interview Question

Q: Answer basic probability and statistics questions

This question evaluates core probability and statistics competencies, including interpretation of Poisson rate parameters from PMF shapes, identification of study bias types (such as survivorship or sampling bias), and properties of covariance and Pearson correlation, framed for a Machine Learning / Data Scientist context.

You are given several short, independent probability and statistics questions similar to those in a data / ML screening test. Answer all sub-questions.

1. Ordering Poisson distributions by their rate parameter

You are told that three different Poisson-distributed random variables $X_A, X_B, X_C$ have their probability mass functions (PMFs) plotted on the same graph (support on non-negative integers). The plots are described as follows:

Distribution A : Most of its probability mass is concentrated on values 0, 1, and 2. The mode (highest bar) is at 1. The 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: there is still 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 $\lambda_A$ , $\lambda_B$ , and $\lambda_C$ respectively.

Question 1: Based on the qualitative description of the plots, order the three rate parameters from smallest to largest:

(a) $\lambda_A$ , $\lambda_B$ , $\lambda_C$
(b) $\lambda_C$ , $\lambda_B$ , $\lambda_A$
(c) $\lambda_A$ , $\lambda_C$ , $\lambda_B$
(d) $\lambda_B$ , $\lambda_A$ , $\lambda_C$

Pick 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 such as team size, prior founder experience, average age of founders, and industry, and then publish a report claiming: “These are the characteristics that make startups successful.”

You are asked: 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 $\text{Cov}(X, Y) = 0$ , then $X$ and $Y$ are independent.
Statement B: The Pearson correlation coefficient $\rho_{XY}$ is always between $-1$ and $1$ , inclusive.
Statement C: The Pearson correlation coefficient between $X$ and $Y$ is given by

$\rho_{XY} = \frac{\text{Cov}(X, Y)}{\sigma_X \sigma_Y},$

where $\sigma_X$ and $\sigma_Y$ are the standard deviations of $X$ and $Y$ , respectively. 4. 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 $Y$ . 5. 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.

You are given several short, independent probability and statistics questions similar to those in a data / ML screening test. Answer all sub-questions.

1. Ordering Poisson distributions by their rate parameter

Distribution A : Most of its probability mass is concentrated on values 0, 1, and 2. The mode (highest bar) is at 1. The 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: there is still 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 $\lambda_A$ , $\lambda_B$ , and $\lambda_C$ respectively.

Question 1: Based on the qualitative description of the plots, order the three rate parameters from smallest to largest:

(a) $\lambda_A$ , $\lambda_B$ , $\lambda_C$
(b) $\lambda_C$ , $\lambda_B$ , $\lambda_A$
(c) $\lambda_A$ , $\lambda_C$ , $\lambda_B$
(d) $\lambda_B$ , $\lambda_A$ , $\lambda_C$

Pick the correct ordering and briefly justify your choice.

2. Identifying type of bias in a study

You are asked: 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 $\text{Cov}(X, Y) = 0$ , then $X$ and $Y$ are independent.
Statement B: The Pearson correlation coefficient $\rho_{XY}$ is always between $-1$ and $1$ , inclusive.
Statement C: The Pearson correlation coefficient between $X$ and $Y$ is given by

$\rho_{XY} = \frac{\text{Cov}(X, Y)}{\sigma_X \sigma_Y},$

State True/False for each and justify in one or two sentences.

Answer basic probability and statistics questions

Quick Overview

1. Ordering Poisson distributions by their rate parameter

2. Identifying type of bias in a study

3. True/False questions on covariance and correlation

Solution

Submit Your Answer

Answer basic probability and statistics questions

Quick Overview

1. Ordering Poisson distributions by their rate parameter

2. Identifying type of bias in a study

3. True/False questions on covariance and correlation

Solution

Submit Your Answer