Bounds: Parameters a and b define the complete support. Constant density: Every equal-width subinterval has equal probability. Decision use: A conservative scenario when only a credible range is known.

f(x)=1/(b-a); P(c<X<d)=(d-c)/(b-a); mu=(a+b)/2; sigma=(b-a)/sqrt(12)

1. گام 1Verify a<b and clip the event to the support.
2. گام 2Divide favorable width by total width.
3. گام 3Explain why equal density is plausible.

متغیر

Define X from the scenario and state its measurement or count units before choosing a model.

انتخاب مدل

Use uniform model only when the support, shape, and process assumptions match the scenario.

پارامترها

Symbols: a and b are support bounds with a<b; c and d are event bounds; mu is the mean; sigma is the standard deviation. Units: a, b, c, d, mu, and sigma use the variable's units.

بیان احتمال

Find the chance of arrival between minutes 14 and 22.

محاسبه

f(x)=1/(b-a); P(c<X<d)=(d-c)/(b-a); mu=(a+b)/2; sigma=(b-a)/sqrt(12). The probability is 8/20=0.40.

تفسیر

The probability is 8/20=0.40.

بررسی معقول‌بودن

Confirm that probabilities lie from 0 to 1, thresholds use the requested units, and the direction of the event matches the business question.

The probability is 8/20=0.40.

متغیر

Define X from the scenario and state its measurement or count units before choosing a model.

انتخاب مدل

Use uniform model only when the support, shape, and process assumptions match the scenario.

پارامترها

Symbols: a and b are support bounds with a<b; c and d are event bounds; mu is the mean; sigma is the standard deviation. Units: a, b, c, d, mu, and sigma use the variable's units.

بیان احتمال

Find P(X>45).

محاسبه

f(x)=1/(b-a); P(c<X<d)=(d-c)/(b-a); mu=(a+b)/2; sigma=(b-a)/sqrt(12). The upper interval is 15/60=0.25.

تفسیر

The upper interval is 15/60=0.25.

بررسی معقول‌بودن

Confirm that probabilities lie from 0 to 1, thresholds use the requested units, and the direction of the event matches the business question.

The upper interval is 15/60=0.25.

متغیر

Define X from the scenario and state its measurement or count units before choosing a model.

انتخاب مدل

Use uniform model only when the support, shape, and process assumptions match the scenario.

پارامترها

Symbols: a and b are support bounds with a<b; c and d are event bounds; mu is the mean; sigma is the standard deviation. Units: a, b, c, d, mu, and sigma use the variable's units.

بیان احتمال

Find its mean and standard deviation.

محاسبه

f(x)=1/(b-a); P(c<X<d)=(d-c)/(b-a); mu=(a+b)/2; sigma=(b-a)/sqrt(12). The mean is 14 days and SD is 12/sqrt(12).

تفسیر

The mean is 14 days and SD is 12/sqrt(12).

بررسی معقول‌بودن

Confirm that probabilities lie from 0 to 1, thresholds use the requested units, and the direction of the event matches the business question.

The mean is 14 days and SD is 12/sqrt(12).

متغیر

Define X from the scenario and state its measurement or count units before choosing a model.

انتخاب مدل

Use uniform model only when the support, shape, and process assumptions match the scenario.

پارامترها

Symbols: a and b are support bounds with a<b; c and d are event bounds; mu is the mean; sigma is the standard deviation. Units: a, b, c, d, mu, and sigma use the variable's units.

بیان احتمال

Decide whether uniform remains credible.

محاسبه

f(x)=1/(b-a); P(c<X<d)=(d-c)/(b-a); mu=(a+b)/2; sigma=(b-a)/sqrt(12). Bounds alone do not justify equal density.

تفسیر

Bounds alone do not justify equal density.

بررسی معقول‌بودن

Confirm that probabilities lie from 0 to 1, thresholds use the requested units, and the direction of the event matches the business question.

Bounds alone do not justify equal density.

dunif() returns density, punif() cumulative probability, qunif() a quantile, and runif() random values. Run the original starter code, inspect every printed intermediate value, then modify one assumption and explain the decision impact. Keep code, formulas, function names, column names, and numerical output left-to-right.

Name the assumption, the evidence you would seek, and how the recommendation could change.

Use bounded equal-density models for transparent interval and what-if calculations. The reliable workflow is: define X and units, justify the model, state the event, calculate, verify, interpret, and disclose limitations.

Use the short exit check after instruction and practice. Reach 70% to complete this lesson; explanations appear only after submission.