Question

Summarize key data distribution concepts including probability mass functions (PMF), probability density functions (PDF), and cumulative distribution functions (CDF). Based on your organization or any organization you are most familiar with, provide an example of a PMF, an example of a PDF, and an example of a CDF, based on the type of data used in the organization. How would you summarize each of these to someone who is not familiar with each of these functions?

Answer 1

For a discrete random variable, probability mass function (PMF) is a function representing probability of that variable at discrete points of the sample space.

For a continuous random variable, probability density function (PDF) is a function representing probability of that variable at any point in the sample space.

Cumulative distribution function (CDF) is a function representing probability of that variable from minus infinity to any desired point . The point may not be in the sample space also. It is used in case of discrete as well as continuous variable.

EXAMPLES-

PMF- Suppose, out of 100 bulbs produced in 4 machines, 30, 25, 23 and 22 bulbs are produced from first, second, third and fourth machines respectively. Random variable X serial number of the machine. Then the PMF is given by,

Clearly, sum of the probabilities is 1.

PDF- Suppose, random variable Y denotes life span of bulbs in days, Then PDF can be as follows.

CDF- The example used in case of PMF can be again used in case of this one. This yields,

Here F(X) calculates the probability of a bulb to be produced by any machine numbered as less than or equal to X.