### Probability Theory Review¶

•  (a) (#) Proof that the "elementary" sum rule $p(A) + p(\bar{A}) = 1$ follows from the (general) sum rule $$p(A+B) = p(A) + p(B) - p(A,B)\,.$$
(b) (###) Conversely, derive the general sum rule $p(A + B) = p(A) + p(B) - p(A,B)$ from the elementary sum rule $p(A) + p(\bar A) = 1$ and the product rule. Here, you may make use of the (Boolean logic) fact that $A + B = \overline {\bar A \bar B }$.
•  Box 1 contains 8 apples and 4 oranges. Box 2 contains 10 apples and 2 oranges. Boxes are chosen with equal probability.
(a) (#) What is the probability of choosing an apple?
(b) (##) If an apple is chosen, what is the probability that it came from box 1?
•  (###) The inhabitants of an island tell the truth one third of the time. They lie with probability $2/3$. On an occasion, after one of them made a statement, you ask another "was that statement true?" and he says "yes". What is the probability that the statement was indeed true?
•  (##) A bag contains one ball, known to be either white or black. A white ball is put in, the bag is shaken, and a ball is drawn out, which proves to be white. What is now the chance of drawing a white ball? (Note that the state of the bag, after the operations, is exactly identical to its state before.)
•  A dark bag contains five red balls and seven green ones.
(a) (#) What is the probability of drawing a red ball on the first draw?
(b) (##) Balls are not returned to the bag after each draw. If you know that on the second draw the ball was a green one, what is now the probability of drawing a red ball on the first draw?
•  (#) Is it more correct to speak about the likelihood of a model (or model parameters) than about the likelihood of an observed data set. And why?
•  (##) Is a speech signal a 'probabilistic' (random) or a deterministic signal?
•  (##) Proof that, for any distribution of $x$ and $y$ and $z=x+y$ \begin{align*} \mathbb{E}[z] &= \mathbb{E}[x] + \mathbb{E}[y] \\ \mathbb{V}[z] &= \mathbb{V}[x] + \mathbb{V}[y] + 2\mathbb{V}[x,y] \end{align*} where $\mathbb{E}[\cdot]$, $\mathbb{V}[\cdot]$ and $\mathbb{V}[\cdot,\cdot]$ refer to the expectation (mean), variance and covariance operators respectively. You may make use of the more general theorem that the mean and variance of any distribution $p(x)$ is processed by a linear tranformation as \begin{align*} \mathbb{E}[Ax +b] &= A\mathbb{E}[x] + b \\ \mathbb{V}[Ax +b] &= A\,\mathbb{V}[x]\,A^T \end{align*}
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