In the previous part, we covered some Probability Distributions (Discrete), today we gonna talk about some Continuous Distributions (The Normal Distribution and The standard Normal Distribution).

Continuous Distribution👇

Continuous distribution has infinitely many consecutive outcomes.

The sample space of Continuous distribution is infinite and we cannot record the frequency of each distinct value.

So we cannot represent them with a table but we can represent them with a graph.

The graph of a Continuous Distribution is called the graph of Probability Density Function(PDF).

We denote it as f(y) where y is an element of sample space.

The f(y) function depicts the associated probability for every possible value ‘y’.

The curve of bars of Continuous Distribution are being called as Probability Distribution Curve(PDC)

The Probability distribution curve shows the likelihood of each outcome.

In continuous, the greater the denominator becomes, the closer the function is to 0.

In continuous, the probability is extremely insignificant on more likely to be 0.

We have been using the term probability function to refer to the probability density function or PDF.

Cumulative Distribution Function(CDF) depicts the function that encompasses everything up to a certain value.

We denote the CDF ‘F(y)’ for any continuous random variable

The CDF is more useful when we want to estimate the probability of some interval. For example-

We find the area by computing the integral of the density curve over The integral A to B.

To calculate integral use some online software like wolframalpha.com

The expected value of the continuous distribution is an integral, Like-

The variance of a continuous distribution is some as discrete distribution.

Continuous Distribution: The Normal Distribution👇

We define a Normal Distribution using “N(µ,σ²)”

The Normal Distribution frequently appears in nature as well as in life, in various shapes and forms. For example-

The size of a fully grown male lion follows a Normal Distribution.

The graph of a Normal Distribution is bell-shaped.

The expected value of a normal distribution is-

Or,

Another peculiarity of the normal distribution is the “ 68.95.99.7 “ law -

The outcomes are extremely rare in a Normal Distribution.

Continuous Distribution: The Standard Normal Distribution👇

Before understanding standardizing we need to know about transformation.

What is Transformation?👇

A transformation is a way in which we can after every distribution get a new distribution with similar characteristics.

If we add a constant to every element of a Normal Distribution the new distribution would still be normal-

If we had a constant like 3 to the entire distribution, we simply need to move the graph 3 places to the right. For example-

If we subtract a number from every element we would simply move our current graph to the left to get the new one-

If we multiply the function by a constant it will shrink that many times -

If we divide every element by a number the graph will expand-

However, If we multiply or divide by a number between zero and one (1<c>0), the opposing effect will occur. For example-

Dividing by ½ is the same as multiplying by 2 so, the graph will shrink even though we are dividing

What is Standardizing?👇

Standardizing is a special kind of transformation in which we make the expected value equal to 0 (E(x) = 0) and the variance equal to 1 (var(x) = 1)

The distribution we get after standardizing any normal distribution is called a standard normal distribution.

We wish to move the graph either to the left or to the right until its mean equals 0. The way we do that is by subtracting the mean from every element

After this, to make the standardization complete we need to make sure the standard deviation is one to do so we have to divide every element of the newly obtained distribution by the value of the standard deviation sigma(σ).

If we denote the standard Normal Distribution with z then for any Normally distributed variable y then -

This equation expresses the transformation we use when standardizing

Every element of the non-standardized distribution is represented in the new distribution by the number of standard deviations it is away from the mean.

In cases where we have less than 30 entries we usually event assuming a normal distribution.

Before we end…

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