How is pdf related to cdf

In technical terms, a probability density function pdf is the derivative of a cumulative distribution function cdf. Furthermore, the area under the curve of a pdf between negative infinity and x is equal to the value of x on the cdf. For an in-depth explanation of the relationship between a pdf and a cdf, along with the proof for why the pdf is the derivative of the cdf, refer to a statistical textbook.

Your email address will not be published. Skip to content Menu. Posted on June 13, March 2, by Zach. Discrete Random Variables A discrete random variable is one which can take on only a countable number of distinct values like 0, 1, 2, 3, 4, 5…, 1 million, etc.

Some examples of discrete random variables include: The number of times a coin lands on tails after being flipped 20 times. The number of times a dice lands on the number 4 after being rolled times. Continuous Random Variables A continuous random variable is one which can take on an infinite number of possible values.

Some examples of continuous random variables include: Height of a person Weight of an animal Time required to run a mile For example, the height of a person could be Probability Density Functions A probability density function pdf tells us the probability that a random variable takes on a certain value. Cumulative Distribution Functions A cumulative distribution function cdf tells us the probability that a random variable takes on a value less than or equal to x.

Cumulative distribution functions have the following properties: The probability that a random variable takes on a value less than the smallest possible value is zero. For example, the probability that a dice lands on a value less than 1 is zero.

The probability that a random variable takes on a value less than or equal to the largest possible value is one. The cdf represents the cumulative values of the pdf. That is, the value of a point on the curve of the cdf represents the area under the curve to the left of that point on the pdf. The total area under the pdf is always equal to 1, or mathematically:. The well-known normal or Gaussian distribution is an example of a probability density function.

The pdf for this distribution is given by:. Again, this is a 2-parameter distribution. Since this function defines the probability of failure by a certain time, we could consider this the unreliability function. Subtracting this probability from 1 will give us the reliability function, one of the most important functions in life data analysis. The reliability function gives the probability of success of a unit undertaking a mission of a given time duration.

The following figure illustrates this. This is the same as the cdf. Reliability and unreliability are the only two events being considered and they are mutually exclusive; hence, the sum of these probabilities is equal to unity. Conditional reliability is the probability of successfully completing another mission following the successful completion of a previous mission. The time of the previous mission and the time for the mission to be undertaken must be taken into account for conditional reliability calculations.

The conditional reliability function is given by:. The failure rate function enables the determination of the number of failures occurring per unit time. Omitting the derivation, the failure rate is mathematically given as:. This gives the instantaneous failure rate, also known as the hazard function. So this result tells us that, to approximate the probability that the random variable lies in a given interval, we just have to guess the fraction of the area under the pdf between the ends of the interval.

This result provides another perspective on why pdfs cannot be negative, since if they were, a negative probability could be obtained, which is impossible. The pdf is analogous to, but different from, the probability function pf for a discrete random variable. A pf gives a probability, so it cannot be greater than one.

On the other hand, the height of the curve reflects the relative probability.