Definition:
The Probability Density Function(PDF) of a continuous random variable is a function which
can be integrated to obtain the probability that the random variable takes a value in a given interval. PDF is used to find
the point of Normal Distribution curve. Continuous Probability Density Function of the Normal Distribution is called the Gaussian Function.
Standard Normal Distribution:
A random variable which has a normal distribution with a mean m=0 and a standard deviation σ=1
is referred to as Standard Normal Distribution.
Formula:
PDF of Normal Distribution = P(x) = (1/(σsqrt(2π)))e-(x-m)2 / (2σ2)
PDF of Standard Normal Distribution = P(x) = (1/sqrt(2π))e-(x2 / 2)
where,
m = Mean.
σ = Standard Deviation.
π = 3.14
e = 2.718
Example 1: Find Probability Density Function with,
mean m=5
Standard deviation σ=2
Normal random variable x=10
Step 1: To calculate PDF find sqrt(2π).
sqrt(2π) = sqrt(2 x 3.14)
= sqrt(6.28) = 2.51
Step 2: Find 1/(σsqrt(2π)).
σsqrt(2π) = 2 x 2.51 = 5.02
1/(σsqrt(2π)) = 1/5.02 = 0.199
Step 3: To Find e-(x-m)2 / (2σ2) calculate -(x-m)2 and 2σ2.
-(x-m)2 = -(10-5)2
= 52 = 25
2σ2 = 2 x (22)
= 2 x 4 = 8
-(x-m)2 / (2σ2) = 25/8
= 3.125
Step 4: Calculate e-(x-m)2 / (2σ2)
= 2.7183.125 = 22.75
Step 5: To find PDF formula is used.
= 0.199 x 22.75 = 4.53
Example 2: Find Standard Normal Distribution(m=0; σ=1) with,
Normal random variable x=2
Step 1: Find 1/sqrt(2π).
sqrt(2π) = 2.51
1/sqrt(2π)) = 1/2.51 = 0.39
Step 2: Calculate e-(x2 / 2).
(x2 / 2)= 22/2 = 2
e-(x2 / 2)= 2.7182 = 7.387524
Step 3: To find Standard Normal Distribution Formula is used.
= 0.39 x 7.387524 = 2.9
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