Continuous random variable: A measurement model whose possible values fill intervals. PDF: A nonnegative density whose total area is one. CDF: F(x)=P(X<=x), the accumulated left-tail area.

P(a<X<b)=F(b)-F(a); P(X=x)=0

1. Step 1Define X and its units.
2. Step 2Translate the event into an interval.
3. Step 3Subtract cumulative areas and interpret.

Variable

X is fill volume in millilitres, with support 5<=X<=15.

Model choice

A constant PDF is valid because its nonnegative rectangular area can be normalized to one.

Parameters

The support width is 15-5=10 mL, so the density height is 1/10=0.1 per mL.

Probability statement

P(8<X<12)

Calculation

Area = width x height = (12-8)(0.1)=0.40.

Interpretation

Under this model, 40% of fills lie between 8 mL and 12 mL.

Reasonableness check

The shaded width is 4 of the 10 supported millilitres, so 0.40 is plausible; the full area is 10(0.1)=1.

The support is 10 mL wide, so height 1/10=0.1 per mL makes total area one. The requested rectangle is 4 mL wide, so P(8<X<12)=4(0.1)=0.40.

Variable

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

Model choice

Use area and cumulative probability only when the support, shape, and process assumptions match the scenario.

Parameters

Symbols: X is the measured variable; f(x) is density per unit; F(x) is cumulative probability; a and b are interval boundaries. Restrictions: f(x)>=0 and total area=1. Units: f(x) has reciprocal units; probability has no units.

Probability statement

Classify X and describe P(2<X<8).

Calculation

P(a<X<b)=F(b)-F(a); P(X=x)=0. Temperature is continuous and the event is an interval area.

Interpretation

Temperature is continuous and the event is an interval area.

Reasonableness check

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

Temperature is continuous and the event is an interval area.

Variable

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

Model choice

Use area and cumulative probability only when the support, shape, and process assumptions match the scenario.

Parameters

Symbols: X is the measured variable; f(x) is density per unit; F(x) is cumulative probability; a and b are interval boundaries. Restrictions: f(x)>=0 and total area=1. Units: f(x) has reciprocal units; probability has no units.

Probability statement

Find P(3<X<7).

Calculation

P(a<X<b)=F(b)-F(a); P(X=x)=0. The interval probability is F(7)-F(3)=0.58.

Interpretation

The interval probability is F(7)-F(3)=0.58.

Reasonableness check

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

The interval probability is F(7)-F(3)=0.58.

density() estimates shape from data; it is not a probability at a point. 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.

Move from count probabilities to measurement models and connect density, cumulative probability, and interval area. 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.