Quant Academy Math · Probability · Quant Finance
Phase 7 Exam

Probability Theory - Phase Exam

20 interleaved questions drawn from every lesson in this phase. Interleaving mixes topics on purpose - that difficulty is what builds durable, transferable understanding. Aim for 70%+ before advancing; below that, revisit the flagged lessons.

How this exam teaches
Questions are shuffled across lessons (not blocked by topic) so you practice choosing the right idea. Missed questions are added to your review queue automatically.
Q1 Medium
Why do we restrict events to a sigma-algebra rather than all subsets of \(\Omega\)?
To save memory in computations
Because for uncountable \(\Omega\) no countably-additive probability can be consistently defined on every subset
Because sigma-algebras are easier to write down
Because the power set is not a set
Q2 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
Q3 Medium
If \(X\) is independent of the sigma-algebra \(\mathcal{G}\), then \(\E[X\mid\mathcal{G}]\) equals:
\(X\)
\(0\)
\(\E[X]\) (a constant)
undefined
Q4 Medium
Why is the characteristic function preferred over the MGF for proving limit theorems?
It is easier to differentiate
It always exists and Levy’s theorem links its convergence to convergence in distribution
It gives exact moments
It is real-valued
Q5 Easy
The weak law of large numbers asserts convergence of \(\bar X_n\) to \(\mu\) in the sense of:
Almost sure convergence
Convergence in probability
Convergence in distribution only
Uniform convergence
Q6 Easy
The correct standardization in the classical CLT divides the centered sum by:
\(n\)
\(\sqrt n\)
\(n^2\)
\(\log n\)
Q7 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
Q8 Medium
The law (distribution) of a random variable \(X\) is:
The function \(X:\Omega\to\R\) itself
The sample space \(\Omega\)
The pushforward measure \(\mu_X(B)=\Prob(X\in B)\) on \(\R\)
The value \(X(\omega)\) for the realized \(\omega\)
Q9 Medium
For a convex function \(\varphi\), Jensen’s inequality states:
\(\varphi(\E X)\ge\E[\varphi(X)]\)
\(\varphi(\E X)\le\E[\varphi(X)]\)
\(\varphi(\E X)=\E[\varphi(X)]\)
No general relation holds
Q10 Easy
The tower property \(\E[\E[X\mid\mathcal{G}]]=\E[X]\) expresses that:
Conditioning changes the mean
Averaging the conditional forecasts recovers the unconditional mean
\(X\) must be \(\mathcal{G}\)-measurable
Conditional expectation is deterministic
Q11 Easy
For independent \(X\sim\)Poisson(2), \(Y\sim\)Poisson(3), the law of \(X+Y\) is:
Poisson(5)
Poisson(6)
Normal(5,5)
Binomial(5,·)
Q12 Medium
In the Chebyshev proof of the WLLN, the key fact is that \(\Var(\bar X_n)\) equals:
\(\sigma^2\)
\(n\sigma^2\)
\(\sigma^2/n\)
\(\sigma^2/\sqrt n\)
Q13 Medium
Which implication between convergence modes is TRUE in general?
In distribution \(\Rightarrow\) almost sure
In probability \(\Rightarrow\) in distribution
In distribution \(\Rightarrow\) in probability
\(L^2\Rightarrow\) almost sure
Q14 Easy
For a martingale \((M_n)\), \(\E[M_{n+1}\mid\F_n]\) equals:
\(M_0\)
\(M_n\)
\(\E[M_n]\)
\(0\)
Q15 Easy
Which statement about a density \(f_X\) is correct?
\(f_X(x)\) is the probability that \(X=x\)
\(f_X(x)\) must be at most 1
\(\int_a^b f_X\,dx=\Prob(a\le X\le b)\)
\(f_X\) equals the CDF
Q16 Medium
A variable has mean 50 and standard deviation 5. The best distribution-free bound on \(\Prob(|X-50|\ge20)\) is:
\(\le 1/4\)
\(\le 1/16\)
\(\le 1/2\)
exactly 0.05
Q17 Hard
Conditional expectation \(\E[X\mid\mathcal{G}]\) is characterized as the \(\mathcal{G}\)-measurable variable that:
Maximizes \(\E[XY]\)
Has the same integral as \(X\) over every set in \(\mathcal{G}\)
Equals \(X\) everywhere
Is always constant
Q18 Medium
The MGF satisfies \(M_X^{(k)}(0)=\) ?
\(\Prob(X=k)\)
\(\E[X^k]\)
\(k!\,\Var(X)\)
\(\phi_X(k)\)
Q19 Hard
For i.i.d. Cauchy random variables the sample mean \(\bar X_n\):
Converges to 0 by the SLLN
Converges to the median
Does not converge; it stays Cauchy-distributed
Converges but slowly
Q20 Hard
The characteristic-function proof of the CLT works because \(\phi_Y(t/\sqrt n)^n\to\):
\(e^{-t^2/2}\)
\(e^{-|t|}\)
\(1\)
\(e^{it}\)
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