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The probability density function (PDF) of a continuous random variable • A random variable X is continuous if P (X = x) = 0 for every x. • PDF (機率密度函數) of a continuous random variable. For a continuous random variable X, if there exists a non-negative function f such that Z b P (a < X < b) = f (x)dx for all a < b, (1) a then f is called the probability density function (PDF) of X. – (1) implies that the distribution of X can also be described using its PDF. Rb – The notation a f (x)dx is called the integral of f from a to b (f 從 a 到 b 的積分), which is the area of region D in the following graph. Example 1. Suppose that X has PDF 0 2x/3 f (x) = (3 − x)/3 0 The graph of f is given below. 1 f , where if if if if x < 0; 0 ≤ x ≤ 1; 1 ≤ x ≤ 3; x > 3. Find P (1 < X < 3). Sol. The area under the PDF curve when 0 < x < 3 is (3 − 1) × (2/3)/2 = 2/3, so P (1 < X < 3) = 2/3. • Suppose that X is a continuous random variable, then the PMF of X is not useful for characterizing the distribution of X. • Suppose that X is a random variable. – If X is discrete, then the distribution of X can be characterized using its PMF. – If X is continuous with PDF f , then the distribution of X can be characterized using f . • If X is continuous with PDF f , then the mean and variance of X can be calculated using Z ∞ E(X) = xf (x)dx −∞ and Z ∞ (x − µ)2 f (x)dx, V ar(X) = −∞ where µ = E(X). Computing E(X) and V ar(X) using the above formulas is beyond the scope of this course. 2