Cumulative Distribution Function

Notes:

• Properties of random variables are determined by their probability distribution. The cumulative distribution function (cdf) provides a very useful way for specifying such distributions.
• Given a random variable X and its cdf, we can find the probability of any event involving X. For instance,
• To obtain the cdf of a discrete (continuous) random variable, we sum (integrate) its pmf (pdf).
• You can obtain the pmf (pdf) of a discrete (continuous) random variable by differencing (differentiating) its cdf.
• The cdf of a discrete random variable X is a non-decreasing step function with jumps of magnitude P(X = u) at values of u in the range of X.
• The cdf of a continuous random variable is, in general, a non-decreasing function over the entire real line. For those usually encountered in practice, the cdf is strictly increasing over the range of the variable.