Frequently Asked Questions on Statistics for Data Scientist Interviews.

Question 1: What is a random variable?

Answer: A random variable is a variable whose possible values are numerical outcomes of a random phenomenon. There are two types of random variables:

Discrete Random Variable: Takes on a countable number of distinct values. Examples include the roll of a die or the number of heads in a series of coin flips.

Continuous Random Variable: Takes on an infinite number of possible values. Examples include the exact height of a person or the time taken to run a marathon.

Random variables are often denoted by capital letters such as $X$ or $Y$, while their specific values are denoted by lowercase letters such as $x$ or $y$.

Question 2: Explain the concept of probability distributions.

Answer: A probability distribution describes how the values of a random variable are distributed. There are two main types:

Discrete Probability Distribution: Lists all the possible values of a discrete random variable and the probabilities associated with each value. An example is the binomial distribution.

Continuous Probability Distribution: Describes the probabilities of a continuous random variable. An example is the normal distribution, where probabilities are given over intervals, not specific values.

The probability distribution for a random variable $X$ can be represented as a probability mass function (PMF) for discrete variables or a probability density function (PDF) for continuous variables.

Question 3: What is the expected value of a random variable?

Answer: The expected value (or mean) of a random variable provides a measure of the central tendency of the distribution of the variable. For a discrete random variable $X$ with possible values $x_1, x_2, \dots, x_n$ and corresponding probabilities $p_1, p_2, \dots, p_n$, the expected value is given by:

$$
E(X) = \sum_{i=1}^{n} x_i \cdot p_i
$$

For a continuous random variable with probability density function $f(x)$, the expected value is:

$$
E(X) = \int_{-\infty}^{\infty} x \cdot f(x) \, dx
$$

The expected value represents the average outcome if the experiment were repeated infinitely many times.

Question 4: What is variance and how is it related to the standard deviation?

Answer: Variance measures the spread of a set of numbers. It is the expected value of the squared deviation of a random variable from its mean. For a random variable $X$, the variance $\sigma^2$ is given by:

$$
\sigma^2 = E[(X - \mu)^2] = \sum_{i=1}^{n} (x_i - \mu)^2 \cdot p_i
$$

where $\mu$ is the mean of $X$.

The standard deviation is the square root of the variance:

$$
\sigma = \sqrt{\sigma^2}
$$

Standard deviation provides a measure of spread in the same units as the variable itself, making it more interpretable.

Question 5: What is a probability density function (PDF)?

Answer: A probability density function (PDF) is a function that describes the likelihood of a continuous random variable taking on a particular value. The PDF, denoted as $f(x)$ for a random variable $X$, must satisfy the following properties:

$f(x) \geq 0$ for all $x$.

The total area under the curve of $f(x)$ is 1:

$$
\int_{-\infty}^{\infty} f(x) \, dx = 1
$$

The probability that $X$ falls within a particular range $[a, b]$ is given by the integral of the PDF over that range:

$$
P(a \leq X \leq b) = \int_{a}^{b} f(x) \, dx
$$

Unlike a probability mass function (PMF) for discrete variables, a PDF does not give probabilities for specific values but rather for intervals.

Question 6: What is the cumulative distribution function (CDF)?

Answer: The cumulative distribution function (CDF) of a random variable $X$ is a function that gives the probability that $X$ will take a value less than or equal to $x$. It is defined as:

$$
F(x) = P(X \leq x)
$$

For a discrete random variable, the CDF is the sum of the probabilities for all outcomes less than or equal to $x$:

$$
F(x) = \sum_{t \leq x} P(X = t)
$$

For a continuous random variable, the CDF is the integral of the PDF from $-\infty$ to $x$:

$$
F(x) = \int_{-\infty}^{x} f(t) \, dt
$$

The CDF is useful for finding probabilities over intervals and understanding the distribution's shape.

Question 7: Explain the law of large numbers.

Answer: The law of large numbers is a fundamental theorem in probability theory that describes the result of performing the same experiment a large number of times. It states that the average of the results obtained from a large number of trials will be close to the expected value, and will tend to become closer as more trials are performed.

There are two forms:

Weak Law of Large Numbers: The sample average converges in probability towards the expected value as the number of trials increases.

Strong Law of Large Numbers: The sample average almost surely converges to the expected value as the number of trials increases.

This law justifies the use of the sample mean as an estimate of the population mean in statistics.

Question 8: What is the central limit theorem?

Answer: The central limit theorem (CLT) states that the distribution of the sum (or average) of a large number of independent, identically distributed random variables approaches a normal (Gaussian) distribution, regardless of the original distribution of the variables, provided the number of variables is sufficiently large.

Mathematically, if $X_1, X_2, \dots, X_n$ are independent and identically distributed random variables with mean $\mu$ and variance $\sigma^2$, then the normalized sum:

$$
\frac{\sum_{i=1}^{n} (X_i - \mu)}{\sigma \sqrt{n}} \to N(0, 1) \text{ as } n \to \infty
$$

approaches a standard normal distribution as $n \to \infty$.

The CLT is crucial because it allows for inference about population parameters using sample statistics even when the population distribution is not normal.

Question 9: What is the mean, and how is it calculated?

Answer: The mean (or average) is a measure of central tendency that represents the sum of all values in a dataset divided by the number of values.

The formula for the mean is:

$Mean = \bar{x} = \frac{\sum_{i=1}^{n} x_i}{n}$

where:

$\bar{x}$ is the mean,

$x_i$ represents each value in the dataset,

$n$ is the number of values in the dataset.

Question 10: What is the median, and how is it determined?

Answer: The median is the middle value in a dataset when the values are arranged in ascending or descending order. It divides the dataset into two equal halves.

If the dataset has an odd number of values, the median is the middle value.

If the dataset has an even number of values, the median is the average of the two middle values.

Question 11: What is the mode, and how does it differ from the mean and median?

Answer: The mode is the value that appears most frequently in a dataset. A dataset can have:

No mode if all values are unique,

One mode if one value appears most frequently,

Multiple modes if several values are equally frequent.

Unlike the mean and median, the mode can be used for both numerical and categorical data.

Question 12: What is the standard deviation, and why is it important?

Answer: The standard deviation measures the amount of variation or dispersion in a dataset. A low standard deviation indicates that the values are close to the mean, while a high standard deviation indicates that the values are spread out over a wider range.

The formula for the standard deviation is:

For a population: $\sigma = \sqrt{\frac{\sum_{i=1}^{n} (x_i - \mu)^2}{n}}$

For a sample: $s = \sqrt{\frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n-1}}$

where: