Recall that continuous random variables have uncountably many possible values (think of intervals of real numbers). Just as for discrete random variables, we can talk about
probabilities for continuous random variables using density functions. The probability density function (pdf), denoted \(f\), of a continuous random variable
\(X\) satisfies the following: The first three conditions in the definition state the properties necessary for a function to be a valid pdf for a continuous random variable. The fourth condition tells us how to use a pdf to calculate probabilities for continuous random variables, which are given by integrals the continuous analog to sums. Example \(\PageIndex{1}\)Let the random variable \(X\) denote the
time a person waits for an elevator to arrive. Suppose the longest one would need to wait for the elevator is 2 minutes, so that the possible values of \(X\) (in minutes) are given by the interval \([0,2]\). A possible pdf for \(X\) is given by
Figure 1: Graph of pdf for \(X\), \(f(x)\) So, if we wish to calculate the probability that a person waits less than 30 seconds (or 0.5 minutes) for the elevator to arrive, then we calculate the following probability using the pdf and the fourth property in Definition 4.1.1: Note that, unlike discrete random variables, continuous random variables have zero point probabilities, i.e., the probability that a continuous random variable equals a single value is always given by 0. Formally, this follows from properties of integrals: Cumulative Distribution Functions (CDFs)Recall
Definition 3.2.2, the definition of the cdf,
which applies to both discrete and continuous random variables. For continuous random variables we can further specify how to calculate the cdf with a formula as follows. Let \(X\) have pdf \(f\), then the cdf \(F\) is given by Relationship between PDF and CDF for a Continuous Random VariableLet \(X\) be a continuous random variable with pdf \(f\) and cdf \(F\).
Example \(\PageIndex{2}\)Continuing in the context of
Example 4.1.1, we find the
corresponding cdf. First, let's find the cdf at two possible values of \(X\), \(x=0.5\) and \(x=1.5\): Figure 2: Graph of cdf in Example 4.1.2 Recall that the graph of the cdf for a discrete random variable is always a step function. Looking at Figure 2 above, we note that the cdf for a continuous random variable is always a continuous function. Percentiles of a DistributionDefinition \(\PageIndex{2}\)The (100p)th percentile (\(0\leq p\leq 1\)) of a probability distribution with cdf \(F\) is the value \(\pi_p\) such that $$F(\pi_p) = P(X\leq \pi_p) = p.\notag$$ To find the percentile \(\pi_p\) of a continuous random variable, which is a possible value of the random variable, we are specifying a cumulative probability \(p\) and solving the following equation for \(\pi_p\): Special Cases: There are a few values of \(p\) for which the corresponding percentile has a special name.
Example \(\PageIndex{3}\)Continuing in the context of Example 4.1.2, we find the median and quartiles.
Is cumulative distribution function same as probability distribution function?The cumulative distribution function is used to describe the probability distribution of random variables. It can be used to describe the probability for a discrete, continuous or mixed variable. It is obtained by summing up the probability density function and getting the cumulative probability for a random variable.
What is the relationship between probability density function and cumulative distribution function?Probability and Random Variables
(1.7), p(x) = F′(x). Thus, the probability density is the derivative of the cumulative distribution function. This in turn implies that the probability density is always nonnegative, p(x) ≥ 0, because F is monotone increasing.
What is the difference between probability and cumulative probability?Probability is the measure of the possibility that a given event will occur. Cumulative probability is the measure of the chance that two or more events will happen. Usually, this consists of events in a sequence, such as flipping "heads" twice in a row on a coin toss, but the events may also be concurrent.
What is the major difference between CDF and PMF or PDF )?The PMF is one way to describe the distribution of a discrete random variable. As we will see later on, PMF cannot be defined for continuous random variables. The cumulative distribution function (CDF) of a random variable is another method to describe the distribution of random variables.
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