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Chapter 2.6
Continuous Random Variable

Knowledge Base > Math> Probability Theory

For continuous variables, it is not possible to write out the entire distribution as a table, because there are infinitely many values. We usually define a probability density function, which satisfies the following properties:

f(x) = lim F(x+-e)--F-(x-)= dF-(x) e→0 e dx f(x)dx = P (x < X ≤ x + dx) f(x) ≥ - - ∞ < x < ∞ ∫ ∞ f(x)dx = 1 ∫ x2 -∞ f(x )dx = P(x1 < X ≤ x2) x1
Note that the f(x) is a density function, not a probability. Thus, it may have value less than 1, or any nonnegative value.

There is a function, called cumulative probability density function F(X), defined as follow:
∫ x F(x) = f(t)dt -∞
Different from the probability density function shown above, this probability distribution function has value in form of probability. Thus, it satisfies the following properties:
0 ≤ F(x) ≤ 1,- ∞ < x < ∞ xl→im- ∞F (x) = 0, lxim→∞ F(x) = 1 F(x) non-decreasing as ,x ↑
If X is a discrete random variable, with values x1,x2,…. Then the cumulative density function at the point xi is:
F(x) = P (X ≤ x) = ∑ P(X = x ) x ≤x i i
If X is a continuous random variable, then there exist a density function f(x) such that,
∫ b F(b)- F (a) = P(a < X ≤ b) = f(x)dx a

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