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Probability distribution function → Discrete random variable
Probability density function → Continuous random variable
A discrete probability distribution that represents a random variable that takes on one of two possible values (1 and 0), with a probability p of being 1 and probability q of being 0.
For example, a coin flip can be modeled with a Bernoulli distribution, where heads is 1 and tails is 0
$$ E(X) = p \\ \sigma^{2} = pq = p(1-p) $$
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P(x success in n trials)
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Discrete distribution where we count only 2 states represented as 1 (for a success) or 0 (for a failure). Binomial distribution represents the probability for x successes in n trials, given a success probability p for each trial.
For example: If the probability of scoring a goal is 0.3 (p), what is chances of scoring 5 (x) goals in 10 (n) tries
$$ P(X=x) = \mathrm{C}_{x}^{n} \times p^{x} \times (1-p)^{n-x} $$
$$ E(X) = np \\ \sigma^2=np(1−p) $$
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P(1st success in k trials)
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A probability distribution that describes the number of trials required to achieve the first success in a sequence of independent and identically distributed Bernoulli trials. It is often used in situations where you repeatedly perform a binary experiment (success or failure) until you achieve the first success.
$$ P(X=k)=(1-p)^{k-1} \times p \\ E(X) = \frac{1}{p} \\ Variance = \frac{1-p}{p^2} $$
where