part 1 Use MATLAB® to calculate the following probabilities and percentiles for the standard normal distribution:
a. ??(??=-2)
b. ??(??=1)
c. ??(-0.4=??=1.6)
d. The 5th percentile
e. The 75th percentile
f. The 99th percentile
part 2
Use MATLAB® to calculate the following probabilities and percentiles for the ??(75,8) distribution:
a. ??(??=70)
b. ??(??=73)
c. ??(75=??=85)
d. The 5th percentile
e. The 60th percentile
f. The 97th percentile
The Correct Answer and Explanation is :
Here is the MATLAB code to compute the required probabilities and percentiles for both the standard normal distribution and the normal distribution with mean 75 and standard deviation 8.
% Part 1: Standard Normal Distribution (mean = 0, std = 1)
mu1 = 0; % Mean of standard normal distribution
sigma1 = 1; % Standard deviation of standard normal distribution
% (a) P(X = -2) - Probability at a single point is always zero for continuous distributions
Pa = normpdf(-2, mu1, sigma1);
% (b) P(X = 1)
Pb = normpdf(1, mu1, sigma1);
% (c) P(-0.4 ≤ X ≤ 1.6)
Pc = normcdf(1.6, mu1, sigma1) - normcdf(-0.4, mu1, sigma1);
% (d) 5th percentile
P5 = norminv(0.05, mu1, sigma1);
% (e) 75th percentile
P75 = norminv(0.75, mu1, sigma1);
% (f) 99th percentile
P99 = norminv(0.99, mu1, sigma1);
% Display results for standard normal distribution
fprintf('Standard Normal Distribution Results:\n');
fprintf('P(X = -2): %f\n', Pa);
fprintf('P(X = 1): %f\n', Pb);
fprintf('P(-0.4 ≤ X ≤ 1.6): %f\n', Pc);
fprintf('5th percentile: %f\n', P5);
fprintf('75th percentile: %f\n', P75);
fprintf('99th percentile: %f\n\n', P99);
% Part 2: Normal Distribution (mean = 75, std = 8)
mu2 = 75; % Mean
sigma2 = 8; % Standard deviation
% (a) P(X = 70)
P70 = normpdf(70, mu2, sigma2);
% (b) P(X = 73)
P73 = normpdf(73, mu2, sigma2);
% (c) P(75 ≤ X ≤ 85)
Pc2 = normcdf(85, mu2, sigma2) - normcdf(75, mu2, sigma2);
% (d) 5th percentile
P5_2 = norminv(0.05, mu2, sigma2);
% (e) 60th percentile
P60_2 = norminv(0.60, mu2, sigma2);
% (f) 97th percentile
P97_2 = norminv(0.97, mu2, sigma2);
% Display results for normal distribution N(75,8)
fprintf('Normal Distribution N(75,8) Results:\n');
fprintf('P(X = 70): %f\n', P70);
fprintf('P(X = 73): %f\n', P73);
fprintf('P(75 ≤ X ≤ 85): %f\n', Pc2);
fprintf('5th percentile: %f\n', P5_2);
fprintf('60th percentile: %f\n', P60_2);
fprintf('97th percentile: %f\n', P97_2);Explanation (300 words)
This MATLAB script calculates probabilities and percentiles for both a standard normal distribution (mean = 0, standard deviation = 1) and a normal distribution with mean 75 and standard deviation 8.
For Part 1, we analyze the standard normal distribution, using normpdf for point probabilities and normcdf for cumulative probabilities. However, for continuous distributions, the probability at a single point (e.g., P(X = -2) or P(X = 1)) is technically zero, though the probability density function (normpdf) provides an estimate of density at that point. To find the probability of an interval, we subtract cumulative distribution function (CDF) values. Percentiles are computed using norminv, which is the inverse CDF function.
For Part 2, we repeat the process for a normal distribution with mean 75 and standard deviation 8. The calculations are analogous to Part 1, but with different mean and standard deviation values. Again, point probabilities are computed using normpdf, while cumulative probabilities for intervals are determined using normcdf. The 5th, 60th, and 97th percentiles are found using norminv.
These computations are useful in statistics and engineering, particularly in hypothesis testing, risk assessment, and quality control. The standard normal distribution serves as a benchmark for many statistical applications, while real-world distributions often involve different means and variances. MATLAB provides efficient built-in functions to handle these computations, making it a valuable tool for probability and statistical analysis.