Question

Define probability density function for normal variable.

Answer 1

Hint: The Probability Density Function (PDF) is the probability function which is represented for the density of a continuous random variable lying between a certain range of values.
It is also called a probability distribution function or just a probability function.

Complete step-by-step solution:
We have to find the definition of probability density function for a normal variable.
In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) can be interpreted as providing a relative likelihood that the value of the random variable would equal that sample.
While the absolute likelihood for a continuous random variable to take on any particular value is 0 (since there are an infinite set of possible values to begin with), the value of the PDF at two different samples can be used to infer, in any particular draw of the random variable, how much more likely it is that the random variable would equal one sample compared to the other sample.
In case of a continuous random variable, the probability taken by X on some given value x is always 0. In this case, if we find \[{\text{P}}({\text{X }} = {\text{x}})\], it does not work. Instead of this, we require to calculate the probability of X lying in an interval (a, b). Now, we have to calculate it for \[{\text{P}}\left( {{\text{a}} < {\text{X}} < {\text{ b}}} \right)\]. This can be done by using a PDF. The Probability distribution function formula is defined as,
\[{\text{P}}\left( {{\text{a}} < {\text{X}} < {\text{b}}} \right) = \int\limits_a^b {f(x)dx} \]

Note: We have to remember that the Probability Density Function (PDF) is the probability function which is represented for the density of a continuous random variable lying between a certain range of values. It is also called a probability distribution function or just a probability function.
The probability density function is non-negative for all the possible values, \[f(x) \geqslant 1\], for all x
The area between the density curve and horizontal X-axis is equal to 1, \[\int\limits_\infty ^\infty {f(x)dx} = 1\]