Continuous Probability Distribution in Statistics


Continuous Probability Distribution


The probabilities of the possible values of a continuous random variable is a continuous distribution.

A continuous random variable is a random variable with a set of possible values i.e. infinite and uncountable.

Example: Height of women in Pune : 60 inch, 60.5 inch, 70.1 inch and so on.

Normal Probability Distribution 

Normal distribution, also known as the Gaussian distribution, is a probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean. In graph form, normal distribution will appear as a bell curve.


This is important to understand if a distribution is normal, there are certain qualities that are consistent and help in quickly understanding the scores within the distribution.

The Normal Distribution has:
  • Mean = Median = Mode
  • Symmetric about the center
  • 50% of values less than mean and 50% greater than mean
Where,
  • P(x) = Normal Probability distribution function
  • x = Normal Random Variable
  • σ = Standard deviation
  • μ = Mean
  • e = Exponential constant = 2.71828
  • π = pi = 3.14 or 22/7
For Example

A company has 500 employees, salary of whom is normally distributed, with an average of Rs.40,000 and Standard deviation of Rs.6000. Suppose you pick a random employee from the 500 employees, what are chances he/she earns less than Rs.30,000.

The following information is available: Distribution: 

Normally distributed Mean: 40,000 SD: Rs.6000


Standard Normal Probability Distribution or Z Score

To find out the answer first of all we need to understand the standard scores or Z score 
  • The number of standard deviations from the mean is also called the "Standard Score", "sigma" or "z-score". Get used to those words!
So to convert a value to a Standard Score ("z-score"):
  • first subtract the observation from the mean: 30,000 – 40,000 = -10,000
  • Then divide by the Standard Deviation: -10,000/6000 = - 1.66
  • And doing that is called "Standardizing"
  • We can take any Normal Distribution and convert it to The Standard Normal Distribution.


Z test

The area under the whole of a normal distribution curve is 1, or 100 percent. 
The ztable helps by telling us what percentage is under the curve at any particular point.
Now we need to use Z table to find the what percentage of is under the curve at any particular time 


Since the z score was negative (-1.66), we need to use the negative Z- Scores From the table, we can find out the probabilities of z score at -1.66 = 0.0485 0r 4.85% It means that when we pick a random employee from the 500 employees, the chances he/she earns less than Rs.30,000 is 4.85%


Instead of using Z test table , we can also find the answer using excel function “normdist”


 = 1-0.95= .05 or 5% or in the excel , you can directly use= 1-normdist(x, mean, standard_dev, cumulative)

  • Even you can find out the chances he/she earns exact Rs.50,000.
  • In this example, you have to use “false” in cumulative instead of “true”.
  • Since the probability of finding exact Rs.50,000 is very low, the answer is around 0%


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