Here is my question:
The size of dust particles was measured in a factory. Pictures of particles were taken with a camera attached to a microscope with 50x magnification. 9 imahes of particles were taken giving the following pixel areas of the interior particles: 14900, 32000, 25000, 38900, 16500, 33400, 30200, 23100, and 28700. Dimensions of one pixel are 46.8x46.8 micrometers. Assuming that dust particles are normally distributed, generate the probability density function of the area in micrometers of dust particles.
Thanks
To generate the probability density function (PDF) of the area of dust particles, we can follow these steps:
Step 1: Calculate the actual area in square micrometers for each pixel area measurement:
To calculate the actual area, we need to multiply the pixel area by the square of the dimension of one pixel (46.8x46.8 micrometers). We get the following actual areas:
14900 * (46.8^2) = 0.1550104 mm²
32000 * (46.8^2) = 0.3315072 mm²
25000 * (46.8^2) = 0.2594376 mm²
38900 * (46.8^2) = 0.4088208 mm²
16500 * (46.8^2) = 0.1729184 mm²
33400 * (46.8^2) = 0.3498648 mm²
30200 * (46.8^2) = 0.3165656 mm²
23100 * (46.8^2) = 0.2414416 mm²
28700 * (46.8^2) = 0.3006036 mm²
Step 2: Determine the mean and standard deviation of the actual area:
To find the mean (μ) of the actual areas, add up all the actual areas and divide by the number of measurements:
μ = (0.1550104 + 0.3315072 + 0.2594376 + 0.4088208 + 0.1729184 + 0.3498648 + 0.3165656 + 0.2414416 + 0.3006036) / 9 = 0.2732200 mm²
To find the standard deviation (σ) of the actual areas, calculate the squared differences between each actual area and the mean, sum them up, divide by the number of measurements, and take the square root:
σ = sqrt((0.1550104 - 0.2732200)² + (0.3315072 - 0.2732200)² + (0.2594376 - 0.2732200)² + (0.4088208 - 0.2732200)² + (0.1729184 - 0.2732200)² + (0.3498648 - 0.2732200)² + (0.3165656 - 0.2732200)² + (0.2414416 - 0.2732200)² + (0.3006036 - 0.2732200)²) / 9 = 0.0803444 mm²
Step 3: Calculate the probability density function (PDF):
Assuming the area of dust particles follows a normal distribution, we can use the probability density function formula:
PDF(x) = (1 / (σ * √(2π))) * e^(-((x - μ)² / (2σ²)))
where PDF(x) is the PDF at a given value x, σ is the standard deviation, μ is the mean, π is a mathematical constant approximately equal to 3.14159, and e is a mathematical constant approximately equal to 2.71828.
Now, let's calculate the PDF for each pixel area measurement using the given values of σ and μ:
For 0.1550104 mm²:
PDF(0.1550104) = (1 / (0.0803444 * √(2π))) * e^(-((0.1550104 - 0.2732200)² / (2 * 0.0803444²)))
For 0.3315072 mm²:
PDF(0.3315072) = (1 / (0.0803444 * √(2π))) * e^(-((0.3315072 - 0.2732200)² / (2 * 0.0803444²)))
Repeat the same calculation for each of the other measurements.
By performing these calculations, you can generate the probability density function (PDF) for the area in micrometers of dust particles.
To generate the probability density function (PDF) of the area of dust particles, we need to perform a series of steps:
1. Convert the given pixel areas to micrometer areas:
To convert the pixel areas to micrometer areas, multiply each pixel area by the square of the dimensions of one pixel. In this case, the dimensions of one pixel are 46.8x46.8 micrometers. So, multiply each pixel area by (46.8 * 46.8) to get the micrometer areas.
The micrometer areas for the given pixel areas are as follows:
14900 * (46.8 * 46.8) = 33044400 micrometer^2
32000 * (46.8 * 46.8) = 71193600 micrometer^2
25000 * (46.8 * 46.8) = 55656000 micrometer^2
38900 * (46.8 * 46.8) = 86300160 micrometer^2
16500 * (46.8 * 46.8) = 36523800 micrometer^2
33400 * (46.8 * 46.8) = 74188800 micrometer^2
30200 * (46.8 * 46.8) = 67092480 micrometer^2
23100 * (46.8 * 46.8) = 51243880 micrometer^2
28700 * (46.8 * 46.8) = 63620640 micrometer^2
2. Calculate the mean (μ) and standard deviation (σ):
To assume that the dust particle sizes are normally distributed, we need to calculate the mean and standard deviation of the micrometer areas.
To calculate the mean, simply sum up all the micrometer areas and divide by the number of measurements (9 in this case):
μ = (33044400 + 71193600 + 55656000 + 86300160 + 36523800 + 74188800 + 67092480 + 51243880 + 63620640) / 9
μ = 60204080 micrometer^2
To calculate the standard deviation, we need to calculate the variance first. The variance can be obtained by summing up the squared differences of each measurement from the mean, and then dividing by the number of measurements (9 in this case):
Variance = [(33044400 - 60204080)^2 + (71193600 - 60204080)^2 + ... + (63620640 - 60204080)^2] / 9
Once we have the variance, the standard deviation can be found by taking the square root of the variance:
σ = sqrt(Variance)
3. Generate the probability density function (PDF):
The PDF can be generated by using the normal distribution formula, which is given by:
PDF(x) = (1 / (σ * sqrt(2 * π))) * e^(-((x - μ)^2 / (2 * σ^2)))
In this formula, x represents the micrometer area of the dust particles.
Using the calculated mean (μ) and standard deviation (σ), plug in the values into the formula for each x to obtain the corresponding probability density.
This is how you generate the probability density function (PDF) of the area in micrometers of the dust particles in the given scenario.