The normal density function, also known as the Gaussian distribution or bell curve, is given by the formula:
f(x) = (1 / σ√(2π)) * e^(-(x-μ)² / (2σ²))
where:
- μ is the mean of the distribution
- σ is the standard deviation of the distribution
- e is the base of the natural logarithm (approximately 2.71828)
Example 1:
Suppose we have a normal distribution with a mean (μ) of 50 and a standard deviation (σ) of 10. Let's calculate the probability density for x = 60.
Substituting the values in the formula:
f(60) = (1 / (10√(2π))) * e^(-(60-50)² / (2(10)²))
Simplifying:
f(60) = (1 / (10 * 2.50663)) * e^(-(10)² / 200)
≈ 0.079577 * e^(-0.5)
≈ 0.079577 * 0.60654
≈ 0.04832
Therefore, the probability density for x = 60 in this normal distribution is approximately 0.04832.
Example 2:
Let's consider another normal distribution with a mean (μ) of 75 and a standard deviation (σ) of 5. We want to find the value of x that has a probability density of 0.03.
Substituting the values in the formula:
0.03 = (1 / (5√(2π))) * e^(-(x-75)² / (2(5)²))
Simplifying:
0.03 = (1 / (5 * 2.50663)) * e^(-(x-75)² / 50)
0.03 * (5 * 2.50663) = e^(-(x-75)² / 50)
0.375399 = e^(-(x-75)² / 50)
Taking the natural logarithm of both sides:
ln(0.375399) = -((x-75)² / 50)
Rearranging the equation:
(x-75)² = -50 * ln(0.375399)
Taking the square root to solve for x:
x - 75 = ± √(-50 * ln(0.375399))
x = 75 ± √(-50 * ln(0.375399))
Since the value inside the square root is negative, there are no real solutions for x in this case.
How do you use normal density function fomulae show with 2 examples?
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