# Normal Distribution

**Topics:**Normal distribution, Standard deviation, Random variable

**Pages:**2 (291 words)

**Published:**March 1, 2011

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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