The interview was conducted over Zoom video conference with 2 professors present. After the introductions, the following problems were asked from my selected topics –

• Probability theory and Discrete Mathematics.

Professor 1 on Probability Theory ( Time: 20 mins including introductions)

• Do you know what is a random variable?
• Can you define it precisely?
• Now suppose you have a biased coin with the probability of a Heads being $p$ and the probability of a Tails being $(1-p)$ . How do you mathematically write the random variable for this?
• Now suppose we are doing an experiment where we toss the same coin $N$ times. What is the sample space of this?
• What do you mean when you say Binomial distribution?
• Can you find the expected value of heads for above problem ($N$ tosses of the given biased coin)?
• Okay, do the derivation.
• Is there any other method to find it?
• Why do you say that you can add expected value of Bernoulli random variables?
• Do the experiments need to be independent (for us to be able to add them to get the expected value)?
• Calculate the expected value via the second method.

Professor 2 on Discrete Maths (Time: 15 mins)

1. You are given $n$ vertices. Each pair of vertices $(v1, v2)$ are connected via an undirected edge. How many edges will be in the graph if self loops are allowed?
1. Suppose there are $n$ vertices and $e$ edges in an undirected graph. Can you say anything about the sum of the degrees of the vertices?
1. Suppose I tell you that $n$ is odd and each vertex has an odd degree, is it possible? Can you write the explanation.
1. Write down $a^2 + b^2 + 2ab$ $<=$  $k*(a+b)^2$ on the paper. Here $a,b$ are non-negative real numbers. Can you say anything about $k$?
1. Now change it to $a^2 + b^2 + 2ab$ $<=$  $k*(a^2+b^2)$ . Can you say anything about $k$?

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