How do you use normal density function fomulae show with 2 examples?
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.
To use the normal density function formulas, follow these steps:
Step 1: Determine the mean (μ) and standard deviation (σ) of the normal distribution.
Step 2: Plug in the values of x, μ, and σ into the formula:
f(x) = (1 / (σ √(2π))) * e^(-(x-μ)^2 / (2σ^2))
Here are two examples to illustrate the calculation:
Example 1: Find the probability density function for a normally distributed variable with a mean (μ) of 60 and a standard deviation (σ) of 5 at x = 65.
Step 1: Given μ = 60 and σ = 5.
Step 2: Substitute the values into the formula:
f(x) = (1 / (5 √(2π))) * e^(-(65-60)^2 / (2(5^2)))
f(x) = (1 / (5 √(2π))) * e^(-25 / 50)
f(x) = (1 / (5 √(2π))) * e^(-0.5)
Example 2: Find the probability density function for a normally distributed variable with a mean (μ) of 75 and a standard deviation (σ) of 10 at x = 80.
Step 1: Given μ = 75 and σ = 10.
Step 2: Substitute the values into the formula:
f(x) = (1 / (10 √(2π))) * e^(-(80-75)^2 / (2(10^2)))
f(x) = (1 / (10 √(2π))) * e^(-25 / 200)
f(x) = (1 / (10 √(2π))) * e^(-0.125)
These examples demonstrate how to use the formula for the normal density function to calculate the probability density at specific values of x.