9 Common Probability Distributions with Mean & Variance derivations

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I have been searching for the derivations (like many of you do too) of the well crammed formulae of Mean & Variance related to various probability distributions. Although, existence of such derivations is there already but in a scattered manner, so I decided to go through each one of them in detail and congregate my interpretation tagging them with much needed derivations of each. The quest for an end to end discussion of random variable probability distributions along with derivations of Mean and Variance for each has come to an end as you read this.

Finally in this blog, you will receive structured details about Random variable types and their different types of probability distributions with relevant examples. Every probability distribution discussed has a following pattern:

4. Mean & Variance derivation to reach well crammed formulae

We will discuss probability distributions with major dissection on the basis of two data types:

We can either plot the probability on count (sum) basis or on proportion basis. E.g.

After tossing a coin 10 times, we can plot the probability of count of heads (0 heads, 1 heads, 2 heads…………10 heads)

After tossing a coin 10 times, we can plot the probability of proportion of heads

We can plot the probability distribution by joining the tip of significantly large number of frequency bins (approaching ∞ which are actually real number values of continuous random variable) using a smooth running curve. The value of the curve defining function f(x) actually depicts the height alone at a particular point, for probability calculation we need to compute area under the curve.

Plotting the height variable of population of males. Assumed range for this example is [150cm, 200 cm] containing all real values i.e. 155.58 cm, 176.2 cm etc.

To calculate the probability of an outcome (real value), we use the concept of integration:

Probability of a value lying between x1 and x2:

2. Two possible outcomes — Success or Failure (Mutually Exclusive and Exhaustive)

If all above features hold for X random variable, then it has a Bernoulli distribution.

2. Two possible outcomes — Success or Failure (Mutually Exclusive and Exhaustive)

5. X represents the number of successes in n trials/events

If all above features hold for X random variable, then it has a Binomial distribution.

2. Two possible outcomes — Success or Failure (Mutually Exclusive and Exhaustive)

5. X represents the number of trials needed to get the first success

So for the first success to occur at xth trial:

- First (x-1) trials must be failures

2. Two possible outcomes — Success or Failure (Mutually Exclusive and Exhaustive)

5. X represents the number of trials needed to get the rth success

So for the rth success to occur at xth trial:

2. Randomly sampling n objects without replacement from a population that contains ‘a’ successes and ‘N-a’ failures

3. X represents the number of successes in a sample

4. Binomial distribution provides a reasonable approximation to the hypergeometric when sampling is done for not more than 5% of the population

1. Counting the number of occurrences of an event in a given unit of time, distance, area, or volume

2. Events occur independently and probability of occurrence in a given length of time does not change through time

3. X represents the number of events in a fixed unit of time

4. Binomial distribution tends toward the Poisson distribution as n ->∞

2. Probability determination using integration (area under the curve)

1. Real number output(continuous) with unequal probability (bell-shaped) of occurrence under the influence of chance causes

2. Probability determination using integration (area under the curve)

2. Measures time per single event (time between events in a poisson process)

4. X represents the time it will take for the successive event to occur

I hope this blog was useful for you and helped you in exploring the genesis of the notorious formulae of probability distributions. With this we end the blog here, I will be posting more in the future…..