Probability

1. Define the conditional probability of $A$ given $B$. Mind singular cases.
2. Annotate each part of the conditional probability formulae.
3. Draw an instance of Venn diagram, depicting the intersection of the events $A$ and $B$. Assume that$A \cup B = H$.

1. What's the probability that it will crash in the next month?
2. What's the probability that it will crash at any given moment?

1. If you randomly pick a chip from the store, what is the probability that it is defective?
2. Suppose you now get two chips coming from the same company, but you don’t know which one. When you test the first chip, it appears to be functioning. What is the probability that the second electronic chip is also good?

1. Given a person is diagnosed positive, what’s the probability that this person actually has the disease?
2. What’s the probability that a person has the disease if two independent tests both come back positive?

In a population in which the ratio of females to males is equal, 5% of females and 0.25% of males have the Dercum disease (Fig. 3.4).

outperform humans in selecting better stock market portfolios, 0.05 of humans and 1 out of 15 monkeys could correctly predict stock market trends correctly.

$P(X) = \frac{2}{9} \\ \\ P(Z) = \frac{7}{9}$

• Distributions
• Expectations
• Variance
• PMFs and CDFs
• Support

• Support
• PMF Equation
• Smooth PMF
• Expectation
• Variance

• Support
• PMF Equation
• Expectation
• Variance

$X ∼ \text{Binomial}(n, p)$

$X ∼ Binomial(n, p)$

• Notation
• Parameters
• Support
• PMF
• Expectation
• Variance

A PT device which was not properly calibrated is used to treat a patient with pancreatic cancer (Fig. 3.1). As a result, a PT beam randomly shoots $200$ particles independently and correctly hits cancerous cells with a probability of $0.1$.

1. Derive the posterior distribution $p(θ|y)$ after observing y.
2. Derive the prior predictive distribution for y.

$x ∼ Ber(γ), γ ∼ B(α,β) ⇒ γ|x ∼ B(α′,β′)$ For instance, for h heads and t tails, the posterior is: $B(h + α,t + β)$

$n(v)dv = \frac{4\pi N}{V}(\frac{m}{2\pi kT})^{\frac{3}{2}}v^2e^{-\frac{mv^2}{2kT}}dv$

• Support
• PDF Expression
• Expectations
• Variance

• Notation
• Parameters
• Support
• Pdf expression
• Cdf expression
• Expectation
• Variance

1. The natural logarithm of the PDF $lnf(X|γ) = Φ(X|γ)$.
2. The first partial derivative $Φ′(X|γ)$.
3. The second partial derivative $Φ′′(X|γ)$.
4. The Fisher Information for a continuous random variable: $I(γ) = −Eγ[Φ′(X|γ)]$ Find the Fisher Information $I(γ)$ for the following distributions:
5. The Bernoulli Distribution $X ∼ B(1, γ)$.
6. The Poisson Distribution $X ∼ Poiss(θ)$.

2. Define the term prior predictive distribution.

1. $Cov(X)$2.$cCov(X)$3.$c^2Cov(X)$4. $0$