Normal Distribution

Introduction

TBA

Density function

The density function for the Normal distribution is $$f(x | \mu, \sigma ) = \frac{1}{\sqrt{2\pi\sigma^2}} e^{-(x - \mu)^2 / (2\sigma^2)}$$

The density function for the Normal distribution depends on the parameters $\mu =$ $\, 1.5 \,$ and $\sigma =$ $\, 3 \,$

Specific to the plot above, the density function is $$f(x | \mu = 1.5, \sigma = 3) = \frac{ 1 } { \sqrt{ 2\pi3^2 } } e^ { -(x - 1.5)^2 / (2 \cdot 3^2) }$$

Examples

1. In a construction project, engineers are using steel rebars for reinforcement. The breaking strength of these rebars is normally distributed with a mean breaking strength of 50,000 pounds per square inch (psi) and a standard deviation of 2,000 psi.
   1. What is the probability that a randomly selected rebar will have a breaking strength of less than 48,000 psi?
   2. To ensure that at least 90% of the rebars in a batch meet a certain strength requirement, what should be the minimum breaking strength specification that engineers should set?
2. Geologists are studying the size of sedimentary rock particles in a particular geological formation. They have found that the size of these particles follows a Normal distribution with a mean (μ) of 10 millimeters and a standard deviation (σ) of 2 millimeters.
   1. What is the probability that a randomly selected sedimentary rock particle from this formation has a size between 8 and 12 millimeters?
   2. What is the probability that a randomly selected sedimentary rock particle from this formation has a size bigger than 14 millimeters?
3. In a physics laboratory, the time it takes for a particle to travel through a specific region follows a Normal distribution with a mean of 5.2 milliseconds and a standard deviation of 0.8 milliseconds.
   1. What is the probability that a randomly selected particle will take less than 4 milliseconds to travel through this region?
   2. Find the probability that a particle will take more than 6 milliseconds to travel through the region.
   3. If the laboratory needs to ensure that at least 90% of particles pass through the region within a certain time, what should be the maximum allowable time for this process?
4. In civil engineering, the compressive strength of concrete is an essential parameter. Suppose the compressive strength of a certain type of concrete follows a Normal distribution with a mean (average) strength of 30 megapascals (MPa) and a standard deviation of 5 MPa. Civil engineers need to ensure that the concrete used in a construction project has a compressive strength of at least 25 MPa to meet safety requirements.
   1. What is the probability that a randomly selected sample of this concrete will have a compressive strength of less than 25 MPa?
   2. If a civil engineering project requires concrete with a minimum compressive strength of 35 MPa, what is the probability that a randomly selected sample will meet this requirement?

Calculator

$X \sim\;$Normal( $\mu =$ $\, 1.5 \,$, $\sigma =$ $\, 3 \,$). $\mathbb{P}[X$ $\le$ $\, 1 \,$ $] = 0.43$.

References

Normal distribution. Wikipedia: The Free Encyclopedia. Wikimedia Foundation, Inc. Accessed 17/10/2023.