A probability density function (PDF) is a mathematical function that describes the relative likelihood of a random variable taking on a given value. It is used to model continuous random variables and can be used to calculate the probability of a variable falling within a certain range of values.
E(g(x)) is the expected value of a function g(x) of a random variable x. It represents the average value of the function over all possible values of x.
Yes, it is possible for two different probability density functions to give the same expected value. This is because the expected value is a measure of central tendency and does not necessarily reflect the shape of the distribution.
To determine if two probability density functions will give the same expected value, you can calculate the expected value for each function and compare the results. If the values are the same, then the functions will give the same expected value.
Knowing if two probability density functions will give the same expected value is important in understanding the behavior of a random variable. It can also be useful in making predictions and decisions based on the expected value of a function.