A normal distribution is a probability distribution in which data is symmetrically distributed around the mean, creating the familiar bell-shaped curve.
Example 1: Find the probability density function of the normal distribution of the following data. x = 2, μ = 3 and σ = 4.
Given,
- Variable (x) = 2
- Mean = 3
- Standard Deviation = 4
Using formula of probability density of normal distribution
f(x,\mu , \sigma ) =\frac{1}{\sigma \sqrt{2\pi }}e^\frac{-(x-\mu)^2}{2\sigma^{2}} Simplifying,
f(2, 3, 4) = 0.09666703
Example 2: If the value of the random variable is 4, the mean is 4, and the standard deviation is 3, then find the probability density function of the Gaussian distribution.
Given,
- Variable (x) = 4
- Mean = 4
- Standard Deviation = 3
Using formula of probability density of normal distribution
f(x,\mu , \sigma ) =\frac{1}{\sigma \sqrt{2\pi }}e^\frac{-(x-\mu)^2}{2\sigma^{2}} Simplifying,
f(4, 4, 3) = 1/(3√2π)e0
f(4, 4, 3) = 0.13301
Practice Problems
Question 1: A normal distribution has a mean of 50 and a standard deviation of 5. What is the probability that a randomly selected value from this distribution is less than 45?
Question 2: If a dataset follows a normal distribution with a mean of 100 and a standard deviation of 15, what is the Z-score for a value of 130? Interpret the Z-score.
Question 3: Given a normal distribution with a mean of 70 and a standard deviation of 10, find the probability that a randomly selected value falls between 60 and 80.
Question 4: In a normally distributed dataset with a mean of 80 and a standard deviation of 10, what value corresponds to the 90th percentile?
Question 5: A sample of 30 students has an average test score of 78 with a standard deviation of 12. Assuming the distribution of test scores is normal, what is the probability that the sample mean score is greater than 82?
