Answer the following questions analytically in Markdown

Prove that the expectation value of the Bernoulli distribution is $p$

$$ E[X] = P(X = 0) \times 0 + P(X = 1)\times 1 = 0 + p = p $$

Prove that the variance of the Bernoulli distribution is $p(1-p)$

$$ \textrm{Var}(X) = E[X^2] - E[X]^2 $$$$ E[X]^2 = p^2 $$$$ E[X^2] = P(X = 0) \times 0^2 + P(X = 1)\times 1^2 = 0 + p = p $$$$ \textrm{Var}(X) = E[X^2] - E[X]^2 = p - p^2 = p(1 - p) $$

Prove that the Bernoulli distribution is normalized.

$$ P(X = 0) + P(X = 0) = p + (1 - p) = 1 $$

Prove that the Poisson distribution is normalized. Hint: remember your sums and series lessons from calculus

$$ P(N = n) = \frac{e^{-\mu}\mu^n}{n!} $$$$ \sum_0^\infty \frac{e^{-\mu}\mu^n}{n!} = e^{-\mu}\sum_0^\infty \frac{\mu^n}{n!} = e^{-\mu} e^\mu = 1 $$

where we have used the Taylor series expansion for the exponential function

Derive an analytical equation for $P(0 < t < T)$ for the exponential distribution. Your input to your function is $T$ and the output is $P(0 < t < T)$.

$$ P(0 < t < T) = \int_0^T \lambda e^{-\lambda t} \,dt = \left. -e^{-\lambda t}\right]_0^T = 1 - e^{-\lambda T} $$