Discrete and continuous distributions, expected value, variance, moment generating functions and transformations.
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- Bernoulli distribution: if p is the probability of success, what are E[X] and Var(X)?
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E[X] = p, Var(X) = p(1 - p)- Binomial distribution B(n, p): write the PMF.
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P(X = k) = C(n,k) * p^k * (1-p)^(n-k), k = 0, 1, ..., n- Binomial distribution B(n, p): what are E[X] and Var(X)?
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E[X] = np, Var(X) = np(1 - p)- Poisson distribution with parameter lambda: write the PMF.
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P(X = k) = (e^(-lambda) * lambda^k) / k!, k = 0, 1, 2, ...- Poisson distribution: what are E[X] and Var(X)?
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E[X] = lambda, Var(X) = lambda- Geometric distribution: if X is the number of trials until the first success, what is E[X]?
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E[X] = 1/p- Geometric distribution (number of trials to first success): what is Var(X)?
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Var(X) = (1 - p) / p^2- What is the memoryless property of the exponential distribution?
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P(X > s + t | X > s) = P(X > t) for all s, t >= 0- Exponential distribution with rate lambda: write the PDF.
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f(x) = lambda * e^(-lambda * x), x >= 0- Exponential distribution: what are E[X] and Var(X)?
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E[X] = 1/lambda, Var(X) = 1/lambda^2- Uniform continuous distribution on (a, b): write the PDF.
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f(x) = 1/(b - a), a < x < b- Uniform continuous distribution on (a, b): what are E[X] and Var(X)?
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E[X] = (a + b)/2, Var(X) = (b - a)^2 / 12- Normal distribution: write the PDF in terms of mean mu and variance sigma^2.
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f(x) = (1 / (sigma * sqrt(2*pi))) * e^(-(x - mu)^2 / (2*sigma^2))- Normal distribution: what are E[X] and Var(X)?
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E[X] = mu, Var(X) = sigma^2- Standard normal distribution: what is the mean and variance?
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Mean = 0, Variance = 1
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