Expectation value of Random Variables, Bernoulli process

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

Notes:

  • Standard deviation looks at how spread out a group of numbers is from the mean, by looking at the square root of the variance.
  • The variance measures the average degree to which each point differs from the mean — the average of all data points.

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.

Bernoulli process

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

Which is

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

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