Cdf, Pdf and P - Value Explained

The probability density function (pdf) and cumulative distribution function (cdf) are two of the most important statistical functions in reliability and are very closely related. When these functions are known, almost any other reliability measure of interest can be derived or obtained.

Figure 1

Figure 2

PDF - Probability Density Function or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) can be interpreted as providing a relative likelihood that the value of the random variable would equal that sample

In a more precise sense , the PDF is used to specify the probability of a random variable falling within a particular range of values , as opposed to taking one value. 

CDF - cumulative distribution function (CDF) of a real-valued random variable X, or just distribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x.

• PDF is actually derivative of cdf
• PDF graph can tell the gradient (intensity) of the slope, in the above figure 2, we can see that at 165 in CDF graph the slope is the highest which can be seen and confirmed in PDF graph, where you can see highest peak at 165.

Figure 3

P - Value -  It is the probability of obtaining test results at least as extreme as the results actually observed.

• Smaller the P-value, Stronger the chances of rejecting Null hypothesis
• Usually P-value of 0.5/5% is consirdered
• P - value < 0.05 , is statistically significant. It indicates that there is less than 5% Probability that Null Hypothesis is correct, hence we can reject it.
• In figure 3, this resultant value lies in the blue shaded region
• P - value > 0.05, is not statistically significant and indicates a strong evidence for Null hypothesis
• Here the value lies in the white region of the figure 3
• P-Value can be calculated by Survival function in python
• scipy.stats.norm.sf(abs()) - for left/right tailed tests
• scipy.stats.norm.sf(abs())*2 for 2 tailed tests , where input should be the Z-Score  

 Survival Function - It is a function that gives the probability that a patient, device or object of interest will survive beyond any specified time.

Figure 4