Continuous random variable:
Continuous in continuous random variable means the value can be any real number between a given range. For example, let a value lie between 1 and 10, then the value can be 1.000001, 6, 2.9, 9.99999, etc.
If it is not continuous, it will be discrete means it will not take any real value between a given range. For example, if we roll a die we will only have values 1,2,3,4,5,6 and no any other values like 1.98, 7.87687 etc.
Random in continuous random variable means, the value can be anything, and we can't predict it.
Now, let's go to probability density function (pdf) and cumulative distribution function (cdf). Both pdf and cdf are the most important functions of statistics, and they are related to each other.
Probability Density Function (PDF):
Probability Density Function (PDF) is probability distribution of continuous random variable.
Let a random variable X, then it will be distributed in the following manner:
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| Fig1. Probability Density Function (PDF) |
Suppose there is a point on the graph:
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| Fig2. Probability Density Function (PDF) with a point |
Based on the above graph we can conclude that the probability of getting 45 is 0.05 i.e. 5%.
Now, suppose we have to find the probability of a random variable x which lies between a and b, see below graph:
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| Fig3. Probability of x which lies between a and b |
Probability of random variable x between a and b will be the area of the graph between a and b, which can be calculated using the formula below, which is integration of the function of x from a to b.
Total area under the curve = 1
Cumulative Distribution Function (CDF):
Cumulative Distribution Function (CDF) is also probability distribution of continuous random variable but it is integration of Probability Density Function means a point on cdf equals to the area under the curve of pdf up to the x-axis of the point.
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| Fig4. Cumulative Distribution Function (CDF) |
Suppose there is a point on the graph, what does this point show? Let's see,
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| Fig5. Cumulative Distribution Function (CDF) with a point |
From the above graph we can conclude that 80% of points are less than 50.8. In the graph we can see that probability is up to 1 because the maximum area under the curve is 1 and 100% points will lie within 70.
Combine both pdf and cdf graphs:
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| Fig6. Graph showing both pdf and cdf |
 and Cumulative Distribution Function (CDF)){kind=link}
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