Question

1. What is the probability density function?

2. In the Business context, can anyone use an example to show how the probability density function is applied in the real world?

Answer 1

1. Probability density function f(x) is a function of probabilities for any continous random variable x. Unlike Discrete probability distributions that expresses probabilities in the form of histograms, the continous probability distributions are smooth curves, where probabilities are expressed as areas under the curves.

a. The vertical coordinate is a function of x, described as f(x) and referred to as
the probability density function.
b. The range of possible x values is along the horizontal axis.
c. The probability that x will take on a value between a and b will be the
area under the curve between points a and b, as shown in below Figure

2. Probability density functions have many business applications. Here I am discussing one scuh application from the exponential distribution.In the context of arrivals at a service counter, x will be a continuous random variable
describing the time between successive arrivals, It follows Exponential distribution.

For x= the length of the interval between occurrences:

pdf for x :
f(x)= for x>0 and >0 where =  the mean and variance
of a Poisson distribution

1/= the mean and standard
deviation of the corresponding
exponential distribution
e= the mathematical constant,
2.71828, the base of the natural logarithm system
Areas beneath the curve:

where k= the time, space, or distance until the next
occurrence

Using this area under the probability density function, we can answer questions like

What is the Probability that the Next arrival will occur at least 5 minutes from now?