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plane, system of linear equations, vector spaces and projections, Inner product, and orthogonal projections, Matrix Decomposition, Eigenvalues and Eigenvectors, vector calculus, linear regression, Dimensionality reduction with PCA, Gaussian Mixtures Models, Density Estimation, Support Vector Machines, Model Implementation in R/Python.
such that 
points uniformly at random from the circumference of a circle. What is the probability that this triangle is acute? Suppose we extend this setup to 
points uniformly at random from the surface area of a sphere. What is the probability that this pyramid/tetrahedron is acute? [All the faces are acute-angled triangles.]
, say the 
like given in the figure. What is the regression line of 
on 
?
unstable molecules in a row, 
. One of the 
pairs of neighbors, chosen at random, combine to form a stable dimer; this process continues until there remain 
isolated molecules no two of which are adjacent.
remains isolated?
.
disjoint regions.
, show that a 
line passes through at least 
of the given 
are two Random Variables with correlation 
What is the correlation between 
such that ![\mathbb{E}[X_k]=k\cdot\mu](https://fractionshub.com/wp-content/ql-cache/quicklatex.com-5a0b0c95fe8a417f076831f9fe7795b0_l3.png "Rendered by QuickLaTeX.com")
for 
and ![\texttt{Var}[X_k]=k](https://fractionshub.com/wp-content/ql-cache/quicklatex.com-e438cd58388f5c39a4b1aeab9904344f_l3.png "Rendered by QuickLaTeX.com")
for 
? Note: Here Best is in the sense of Minimized Mean Squared Error.
simulate a uniformly random point inside it. Note: Uniformly Random on the Area.
of such dimensions such that the product 
is defined. The computer can multiply only two matrices at a time and in whatever order the product 
to find the median of the array obtained by merging the two given arrays.
storey building from which if an egg is dropped, breaks. An unbroken egg can be, of course, repeatedly used. What is the most efficient algorithm for the person to determine the minimum required floor?
) or a radial kernel will outperform a support vector classifier on the training data. Which technique performs best on the test data? Make plots and report training and test error rates in order to back up your assertions
and standard deviation 
denote the correlation between the 
th and 
th observations, then the quantity 
is proportional to the squared Euclidean distance between the