Normal Distribution

Density Function

The density function of the (family of) Normal distribution(s) is

The family depends on the parameters . For each new values of , there is a new member of the family and hence a new density function.

Choose values for and :

Examples

  1. In a construction project, engineers use 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.

    a. What is the probability that a randomly selected rebar will have a breaking strength of less than 48,000 psi?

    b. 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.

    a. What is the probability that a randomly selected sedimentary rock particle from this formation has a size between 8 and 12 millimeters?

    b. 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.

    a. What is the probability that a randomly selected particle will take less than 4 milliseconds to travel through this region?

    b. Find the probability that a particle will take more than 6 milliseconds to travel through the region.

    c. 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 strength of 30 megapascals (MPa) and a standard deviation of 5 MPa.

    a. What is the probability that a randomly selected sample of this concrete will have a compressive strength of less than 25 MPa?

    b. 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?

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References