Given a continuous random variable x has a probability density function (pdf), p(x), in a range of [a, b], then the expectation value (or average) of function g(x) is given by
Since the denominator in the above equation is the cumulative distribution function (cdf) of the given p(x) probability density function (pdf) and it is defined as normalized to 1 so the above equation can be written as
Also for a discrete random variable, expectation formulation for g(x) is:
This means that the integration operators are replaced with summation operators when working with discrete random variables.
Now, if we set g(x) equal to x, i.e., the random variable itself, then we obtain the expectation value (“true mean”) of the random variable as
Again, for a discrete random variable, the expectation value formulation reduces to
The variance formulation of the x ( true variance ) is as follows
The square root of the variance is designated by σ, and referred to as the “standard deviation” that is an indication of how a random variable is distributed about its mean. The “true mean” and “true variance” also are referred to as the population parameters, because they are obtained based on a known probability density function, i.e., population.
A Bernoulli process refers to a physical process that has only two outcomes and the probabilities of these outcomes remain constant throughout the experimentation. The probability density function of a Bernoulli process with outcomes (random variables) n is given by
where p varies in a range of [0,1].
Examples for a Bernoulli process can be coin toss and transmission of particles through a shield.
The expectation value of a Bernoulli R.V. (n) is given by
Remember it is calculated for only one event.
And the variance of this random variable only for one event is given by
Now consider a Bernoulli process is repeated N times, with outcomes n(i), then the sum of these outcomes is
Which itself is another random variable with specific pdf that is binomial distribution.
The distribution function for the probability of obtaining n outcomes (“successes”) out of N experiments (trials) follows a binomial distribution.
What is interested is the expectation value of number of successes (n) which is given by
And the variance of random variable n is given by