Worked solution · Exam P
This Exam P sample reference tests Continuous Distributions. Normalization gives constant 12.5. Integrating the density from zero to six yields 0.46875, which matches choice C.
Independent solution
Setup
Determine the density constant C by requiring the density to integrate to one over its support.
Model
Evaluating the normalization integral gives C=12.5.
Compute
Integrate the normalized density from zero to six; the resulting probability is 0.46875.
Answer
The probability that X is below six rounds to 0.47, corresponding to choice C.
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Original practice · fully worked
A response time X is supported on 0<x<2 and has density proportional to x+1. Find the probability that X is less than 1.
The unnormalized mass below one is 1.5, while total mass is 4. Their ratio is 0.375, choice E.
Setup
Write the density as c(x+1) on the stated support.
Model
Normalize by integrating x+1 from zero to two; the unnormalized total mass is 4.
Compute
The unnormalized mass below one is 1.5, so its share of the total is 1.5/4=0.375.
Answer
Therefore Pr(X<1)=0.375, which is choice E.
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