If there are several integers between 0 and i, they have positive probability values. so P(X<i) > 0.songoku said:
No. Notice that your diagram only has one i < N, but there might be several others. Also, the sum of the probabilities must equal 1, so your diagram is missing a lot of probability.songoku said:
Ah, I see. Now I understand the questionFactChecker said:
A cumulative distribution function (CDF) is a function that describes the probability that a random variable takes on a value less than or equal to a specific value. It is a fundamental concept in probability theory and statistics, providing a complete description of the distribution of a random variable.
The CDF is the integral of the probability density function (PDF) for continuous random variables. Specifically, if \( F(x) \) is the CDF and \( f(x) \) is the PDF, then \( F(x) = \int_{-\infty}^{x} f(t) \, dt \). For discrete random variables, the CDF is the sum of the probabilities of the individual outcomes up to a certain value.
A CDF has several important properties: it is non-decreasing, right-continuous, and ranges from 0 to 1. Mathematically, for any two values \( x_1 \) and \( x_2 \) where \( x_1 \leq x_2 \), \( F(x_1) \leq F(x_2) \). Additionally, \( \lim_{x \to -\infty} F(x) = 0 \) and \( \lim_{x \to \infty} F(x) = 1 \).
To find the probability that a random variable \( X \) falls within a specific range \([a, b]\), you can use the CDF as follows: \( P(a \leq X \leq b) = F(b) - F(a) \). This utilizes the property that the CDF provides the cumulative probability up to any point.
The inverse of the cumulative distribution function, also known as the quantile function, is a function that provides the value below which a given percentage of observations fall. If \( F \) is the CDF, then the inverse CDF \( F^{-1}(p) \) gives the value \( x \) such that \( F(x) = p \). This is useful in various applications, including statistical sampling and hypothesis testing.