# Conditional cdf and pdf relationship

### Conditional Distributions for Continuous Random Variables | STAT /

Joint and Marginal Distributions: Suppose the random variables X and Y have joint probability density function (pdf) fX,Y(x,y). The value of the cumulative. Know basic probability relations (module U5). 2. Define random Objectives: 1. Use and relate joint, marginal, and conditional pdfs and cdfs. In probability theory and statistics, given two jointly distributed random variables X and Y, the this is defined only for non-zero (hence strictly positive) P (X = x). {\displaystyle P(X=x).} P(X=x). The relation with the probability distribution of X given Y is.

So what you see right over here, this is a two-way table. And you can also view this as a joint distribution along these two dimensions. So there's all sorts of interesting things that we could try to glean from this, but what we're going to focus on this video is two more types of distributions other than the joint distribution that we see in this data.

One type is a marginal distribution.

## Marginal and conditional distributions

And a marginal distribution is just focusing on one of these dimensions. And one way to think about it is you can determine it by looking at the margin. So, for example, if you wanted to figure out the marginal distribution of the percent correct, what you could do is look at the total of these rows.

So these counts right over here give you the marginal distribution of the percent correct. Now, a marginal distribution could be represented as counts or as percentages. So if you represent it as percentages, you would divide each of these counts by the total, which is So this right over here in terms of percentages gives you the marginal distribution of the percent correct based on these buckets. Now, you could also think about marginal distributions the other way.

You could think about the marginal distribution for the time studied in the class. Then you would look at these counts right over here. You would say a total of 14 students studied between zero and 20 minutes. You're not thinking about the percent correct anymore.

A total of 30 studied between 21 and 40 minutes. And likewise, you could write these as percentages. Again, this is a 2-parameter distribution. Reliability Function The reliability function can be derived using the previous definition of the cumulative distribution function.

### Marginal and conditional distributions (video) | Khan Academy

From our definition of the cdf, the probability of an event occurring by time is given by: Or, one could equate this event to the probability of a unit failing by time. 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.

## Conditional probability distribution

To show this mathematically, we first define the unreliability function,which is the probability of failure, or the probability that our time-to-failure is in the region of 0 and.

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 Function Conditional reliability is the probability of successfully completing another mission following the successful completion of a previous mission.

### Conditional probability distribution - Wikipedia

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: Failure Rate Function 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. It is useful in characterizing the failure behavior of a component, determining maintenance crew allocation, planning for spares provisioning, etc.

Failure rate is denoted as failures per unit time. Mean Life MTTF The mean life function, which provides a measure of the average time of operation to failure, is given by: The MTTF, even though an index of reliability performance, does not give any information on the failure distribution of the component in question when dealing with most lifetime distributions.

Because vastly different distributions can have identical means, it is unwise to use the MTTF as the sole measure of the reliability of a component. Median Life Median life,is the value of the random variable that has exactly one-half of the area under the pdf to its left and one-half to its right.

It represents the centroid of the distribution. The median is obtained by solving the following equation for.