Practice: Week 0: Assignment 0

A juggler, before going for his show, was looking for red balls among his bags filled with different colored balls. He found that 80% of the bags contain 10 red balls, 10% contain 15 red balls, 5% contain 3 red balls, 3% contain 1 red ball, and 2% contain 0 red balls. What is the mean number of red balls in his bags?

In a mixture of two equally weighted Gaussian distributions, with mean µ1 = 0, µ2 = 2, standard deviation σ1 = 0.5, σ2 = 0.2, find its median.

What is the median and mode of a Poisson Distribution with parameter λ = 3 ?

If the first order moment of a distribution is µ and the second order moment about the origin is given by µ′, then the variance of the distribution is given by

An experiment consisting of n = 20 trials with success probability p = 0.3 is modelled as a binomial distribution. The probability that a binomial random variable equals its expected value is

What is the output of the following code block?

i=0
while i<3:
	print i
	i=i+1
else:
	print 0

What is the output when n=5?

def func(n):
	if n==1:
		return n
	else:
		return n*func(n-1)

What is the output of the following (Python) code?

i=5
while i<=10:
	print("Infinite loop")
	i=i+1

1. Give a big-O estimate for f(n)=(n+1)log(n2+1)+3n2.

Is the following claim True or False: "2n is O(3n) and 3n is O(2n)″.

The solution of the recurrence relation an=an−1 for n≥1 and a0=2 is given as in asymptotic notation

If in an iterative algorithm, the size of data gets halved at every step from the previous one then the number of iterations is

Every real matrix always has an eigenvalue decomposition. True or False?

The diagonal entries of the adjacency matrix of a simple graph (no self-loops) are always

What is the sum of entries in a column of the incidence matrix for an undirected graph, where rows represent vertices?

A full binary tree with n internal nodes has how many total nodes?

For an d-dimensional identity matrix, the trace is equal to:

In the directed graph below, given one performs a Depth First Search (DFS) graph traversal with the starting node as 1, determine the first detected cycle. Assume the order of visit is from lower to higher numbered nodes, i.e., starting from node 3, node 1 will be explored before node 4.

Answer True or False For any random variables X and Y, 𝔼(X+Y)=𝔼(X)+𝔼(Y)

Consider a random variable X with mean μ=2 and variance σ2=3. Compute Var(3X+20) and Var(3X+50)

State True or False. If the covariance between two random variables is 0, they are independent

The Variance of a random variable is always

The random variable X−24 has mean 0 and variance 1. What are the mean and variance of X, respectively?