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1.  **(20 points)**
    \c{Derive the VC dimension of the following classifiers.}

2.

3.  **(20 points)**
    \c{Let $P (x|C)$ denote a Bernoulli density function for a class $C \in {C_1, C_2}$
    and $P (C)$ denote the prior}

    a. \c{Given the priors $P (C_1)$ and $P (C_2)$, and the Bernoulli densities
    specified by $p_1 \equiv p(x = 0|C_1)$ and $p_2 \equiv p(x = 0|C_2)$, derive the
    classification rules for classifying a sample $x$ into $C_1$ and $C_2$ based on the
    posteriors $P (C_1|x)$ and $P (C_2|x)$. (Hint: give rules for classifying $x = 0$ and
    $x = 1$.)}

        For $x=0$, the posteriors $P(C_i | x)$ are given by $P(C_i | x = 0) = \frac{p(x = 0 | C_i) p(C_i)}{p(x = 0)}$.

        - $p(x = 0 | C_i)$ is given to us as $p_1$