Frequently Asked Questions on Probability for Data Scientist Interviews.

Question 1: What is the difference between probability and likelihood?

Answer:

• Probability refers to the measure of the likelihood that an event will occur. It is a value between 0 and 1, where 0 indicates impossibility, and 1 indicates certainty. For a discrete random variable $X$, the probability of a specific outcome $x$ is denoted as $P(X=x)$.

• Likelihood is a concept used in statistical inference. It measures the support provided by the data for each possible value of the parameter. For a given parameter $heta$ and observed data $x$, the likelihood is $L(\theta |x)$, which is often expressed as the probability of the observed data given the parameter, $P(x|\theta )$.

Question 2: Explain Bayes' Theorem.

Answer: Bayes' Theorem is a fundamental concept in probability theory and statistics that describes the probability of an event based on prior knowledge of related conditions. It is expressed as:

$$P(A|B)=\frac{P(B|A)\cdot P(A)}{P(B)}$$

Where:

• $P(A|B)$ is the posterior probability of event $A$ given event $B$.

• $P(B|A)$ is the likelihood of event $B$ given event $A$.

• $P(A)$ is the prior probability of event $A$.

• $P(B)$ is the marginal probability of event $B$.

Question 3: What are independent and mutually exclusive events?

Answer:

• Independent Events: Two events are independent if the occurrence of one does not affect the probability of the other. Mathematically, $P(A\cap B)=P(A)\cdot P(B)$.

• Mutually Exclusive Events: Two events are mutually exclusive if they cannot occur simultaneously. For mutually exclusive events, $P(A\cap B)=0$.

Question 4: Define conditional probability.

Answer: Conditional probability is the probability of an event occurring given that another event has already occurred. It is denoted as $P(A|B)$ and is calculated using the formula:

$$P(A|B)=\frac{P(A\cap B)}{P(B)}$$

provided that $P(B)>0$.

Question 5: What is a random variable?

Answer: A random variable is a variable that takes on different values based on the outcomes of a random experiment. There are two types of random variables:

• Discrete Random Variable: Takes on a finite or countable number of possible outcomes. For example, the roll of a die.

• Continuous Random Variable: Takes on an infinite number of possible values within a given range. For example, the height of people.

Question 6: Explain 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 (i.i.d.) random variables approaches a normal distribution, regardless of the original distribution of the variables. Formally, if $X_1,X_2,\dots ,X_n$ are i.i.d. random variables with mean $\mu$ and variance ${\sigma }^2$, then the standardized sum:

$$Z=\frac{\sum _{i=1}^n{X}_i-n\mu }{\sqrt{n}\sigma }$$

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

Question 7: What is the Law of Large Numbers?

Answer: The Law of Large Numbers (LLN) states that as the size of a sample increases, the sample mean will get closer to the expected value (mean) of the population from which the sample is drawn. Formally, if $X_1,X_2,\dots ,X_n$ are i.i.d. random variables with expected value $E(X_i)=\mu$, then:

$$\underset{n\to \mathrm{\infty }}{lim}\frac{1}{n}\sum _{i=1}^n{X}_i=\mu$$

Question 8: What is a probability distribution?

Answer: A probability distribution describes how the values of a random variable are distributed. It provides the probabilities of occurrence of different possible outcomes. There are two types of probability distributions:

• Discrete Probability Distribution: For discrete random variables, e.g., Binomial distribution.

• Continuous Probability Distribution: For continuous random variables, e.g., Normal distribution.

Question 9: Explain the difference between a probability density function (PDF) and a cumulative distribution function (CDF).

Answer:

• Probability Density Function (PDF): For a continuous random variable, the PDF describes the likelihood of the random variable taking on a specific value. The total area under the PDF curve is 1.

• Cumulative Distribution Function (CDF): The CDF represents the probability that a random variable will take a value less than or equal to a specific value. It is expressed as:
  $$F(x)=P(X\le x)$$

for a random variable $X$.

To Be Continued in Part 2...

In the next part, we will cover frequently asked questions on statistics, including concepts like hypothesis testing, confidence intervals, regression analysis, and more. Stay tuned!

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