Sigman 1 Review of Probability
نویسنده
چکیده
1 Review of Probability Random variables are denoted by X, Y , Z, etc. The cumulative distribution function (c.d.f.) of a random variable X is denoted by F (x) = P (X ≤ x), −∞ < x < ∞, and if the random variable is continuous then its probability density function is denoted by f(x) which is related to F (x) via f(x) = F ′(x) = d dx F (x) F (x) = ∫ x −∞ f(y)dy. The probability mass function (p.m.f.) of a discrete random variable is given by p(k) = P (X = k), −∞ < k <∞, for integers k. 1− F (x) = P (X > x) is called the tail of X and is denoted by F (x) = 1− F (x). Whereas F (x) increases to 1 as x→∞, and decreases to 0 as x→ −∞, the tail F (x) decreases to 0 as x→∞ and increases to 1 as x→ −∞. If a r.v. X has a certain distribution with c.d.f. F (x) = P (X ≤ x), then we write, for simplicity of expression, X ∼ F. (1)
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