Normal Distribution

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Normal Distribution:- A continuous random variable X is a normal distribution with the parameters mean and variance then the probability function can be written as f(x) =   - < x < , - < μ < ,  σ > 0.

When σ2 = 1, μ = 0 is called as standard normal.
Normal distribution problems and solutions – Formulas:
X < μ = 0.5 – Z
X > μ = 0.5 + Z
X = μ = 0.5
where,
μ = mean
σ = standard deviation
X = normal random variable
Normal Distribution Problems and Solutions – Example Problems: Example 1:
If X is a normal random variable with mean and standard deviation calculate the probability of P(X<50). When mean μ = 41 and standard deviation = 6.5 Solution:
Given
Mean μ = 41
Standard deviation σ = 6.5
Using the formula
Z =
Given value for X = 50
Z =
=
= 1.38
Z = 1.38
Using the Z table, we determine the Z value = 1.38
Z = 1.38 = 0.4162
If X is greater than μ then we use this formula
X > μ = 0.5 + Z
50 > 41 = 0.5 + 0.4162
P(X) = 0.5 + 0.4162
= 0.9162
Example 2:
If X is a normal random variable with mean and standard deviation calculate the probability of P(X< 37). When mean μ = 20 and standard deviation = 15 Solution:
Given
Mean μ = 20
Standard deviation σ = 15
Using the formula
Z =
Given value for X = 37
Z =
=
= 1.13
Z = 1.13
Using the Z table, we determine the Z value = 1.13
Z = 1.13 = 0.4332
If X is greater than μ then we use this formula
X > μ = 0.5 + Z **** 37 > 20 = 0.5 + 0.3708 **** P(X) = 0.5 + 0.3708 = 0.8708
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