Cumulative Assessment

Cumulative II - Analysis, Probability & Stochastics (Phases 5–9)

25 interleaved questions spanning multiple phases. Cumulative review is one of the most powerful retention tools - it forces retrieval of older material right when it is starting to fade.

Q1 Medium

\(\int_0^T W_s\,dW_s\) equals:

\(\tfrac12 W_T^2\)

\(\tfrac12 W_T^2-\tfrac12 T\)

\(W_T^2-T\)

\(\tfrac12 W_T^2+\tfrac12 T\)

Q2 Easy

In the Itô multiplication table, \((dW)^2\) equals:

\(0\)

\(dt\)

\(dW\)

\((dt)^2\)

Q3 Medium

Brownian sample paths are:

Smooth and differentiable

Continuous everywhere but differentiable nowhere

Discontinuous with jumps

Piecewise linear

Q4 Easy

A failed Cholesky (a non-positive radicand) on an estimated covariance most likely means:

The code has a bug for sure

The matrix is not positive definite - often too few samples or a bad estimate

The matrix is too large

You must switch to LU

Q5 Easy

In \(PA=LU\) with partial pivoting, the matrix \(L\) is:

Upper-triangular with unit diagonal

Unit lower-triangular with entries \(|\ell_{ik}|\le1\)

Orthogonal

Diagonal

Q6 Easy

For the simple symmetric random walk, \(\mathrm{sd}(S_n)\) grows like:

\(n\)

\(\sqrt n\)

\(\log n\)

constant

Q7 Medium

Fubini’s theorem (as opposed to Tonelli) additionally requires:

\(f\ge0\)

\(f\in L^1(\mu\times\nu)\)

A finite measure space

Continuity of \(f\)

Q8 Hard

Which property guarantees the n-step matrix \(P^n\) actually converges (not just that \(\pi\) exists)?

Recurrence alone

Aperiodicity (together with irreducibility)

A symmetric \(P\)

Having two states

Q9 Medium

For standard Brownian motion, \(\Cov(B_s,B_t)\) with \(s\le t\) equals:

\(st\)

\(\min(s,t)=s\)

\(t-s\)

\(0\)

Q10 Medium

The single algorithmic difference between classical and modified Gram-Schmidt is:

MGS normalizes twice

MGS subtracts projections against the running, partially-orthogonalized vector rather than the original column

MGS uses a different \(R\)

MGS requires pivoting

Q11 Medium

Countable additivity of a measure directly implies all of the following EXCEPT:

Monotonicity

Continuity from below

That every set is measurable

Countable subadditivity

Q12 Medium

Least squares via \(Rx=Q^\top b\) is preferred over the normal equations mainly because:

It uses fewer flops

It keeps the conditioning at \(\kappa(A)\) instead of \(\kappa(A)^2\)

It works for singular \(A\)

It avoids computing \(Q\)

Q13 Hard

The characteristic-function proof of the CLT works because \(\phi_Y(t/\sqrt n)^n\to\):

\(e^{-t^2/2}\)

\(e^{-|t|}\)

\(1\)

\(e^{it}\)

Q14 Medium

To draw \(X\sim\Normal(0,\Sigma)\) from standard normals \(Z\) with \(\Sigma=LL^\top\), set:

\(X=\Sigma Z\)

\(X=LZ\)

\(X=L^{-1}Z\)

\(X=Z/L\)

Q15 Medium

A Gaussian process is completely determined by:

Its sample paths

Its mean function and covariance function

Its maximum over time

Its first four moments

Q16 Easy

In Euler–Maruyama, the noise increment is:

\(\sigma S\,\Delta t\,Z\)

\(\sigma S\,\sqrt{\Delta t}\,Z\)

\(\sigma S\,Z\)

\(\sigma S\,(\Delta t)^2 Z\)

Q17 Medium

The stationary distribution \(\pi\) of a chain with transition matrix \(P\) satisfies:

\(P\pi=\pi\)

\(\pi P=\pi\) with \(\sum_i\pi_i=1\)

\(\pi P=0\)

\(P^2=P\)

Q18 Easy

Markov’s inequality \(\Prob(X\ge a)\le\E[X]/a\) requires which assumption?

\(X\) is normal

\(X\ge 0\)

\(X\) has finite variance

\(X\) is discrete

Q19 Easy

The essential difference between the Lebesgue and Riemann integrals is that Lebesgue:

Partitions the range (y-axis) and weights level sets by measure

Partitions the domain into finer subintervals

Only works for continuous functions

Always gives a larger value

Q20 Hard

The martingale-transform theorem implies that a predictable betting strategy on a fair game yields wealth with:

Positive expected gain

Zero expected gain (still a martingale)

Negative expected gain

Undefined expectation

Q21 Medium

A continuous function on a compact set is guaranteed to:

Be differentiable

Attain its maximum and minimum

Be linear

Have no zeros

Q22 Medium

The Cantor set is an example of a set that is:

Countable and of positive measure

Uncountable and of measure zero

Finite

Non-measurable

Q23 Medium

Which random time is a stopping time with respect to the natural filtration of a price process?

The time of the global maximum over all n

The first time the price reaches 120

The time two steps before the crash

The last time before a peak

Q24 Medium

Hölder’s inequality at \(p=q=2\) specializes to:

Minkowski’s inequality

The triangle inequality

Cauchy–Schwarz

Jensen’s inequality

Q25 Easy

For a martingale \((M_n)\), \(\E[M_{n+1}\mid\F_n]\) equals:

\(M_0\)

\(M_n\)

\(\E[M_n]\)

\(0\)

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