Exponential random variables — Question 214
Original QCE Vault practice · 5 marks
Q214 · Practice questionTechnology-freeComplex familiar5 marks
QUESTION 214 (5 marks)
An exponential waiting time W has 90th percentile exactly 12 minutes.
a)Determine its rate and mean exactly.
b)Determine exactly the probability of waiting more than 6 minutes.
WORKED SOLUTION
Practice marking scheme
5 marksANSWER(a) λ=ln10/12; mean 12/ln10. (b) 1/10. Worked solution
(a) The percentile condition gives e−12λ=0.1. Taking logarithms gives rate ln(10)/12. The mean is its reciprocal. (b) P(W>6)=e−6λ=e−ln10/2=10−1/2. Halving time does not halve the probability; exponential survival is multiplicative. Equivalent mathematically justified methods accepted.
Exact answers unless a decimal accuracy is specified.
Part a: Convert percentile to survival 0.1.
Part a: Solve for the rate.
Part b: Evaluate survival at 6.
Part b: Simplify the exact probability.
Practice question aligned to the current QCAA syllabus; review the worked solution and marking criteria.
View the QCAA syllabusHow many marks did you earn?Compare your working with the guide above.
Related questions
- Q212 · Original practice · 5 marks
Two independent sensors have exponential lifetimes, each with mean 5 hours.
- Q213 · Original practice · 6 marks
Independent components A and B have exponential lifetimes with rates 0.2 and 0.3 per hour. Let T be the first failure time.
- Q215 · Original practice · 8 marks
A survival model, with t in hours, is
S(t)=P(T>t)={e−0.2t,e−0.6−0.4(t−3),0≤t≤3,t>3. - Q251 · Original practice · 12 marks
Each round starts two independent unlocking processes. Their completion times T1 and T2 are exponential with means 2 and 3 minutes respectively. The round ends at T=min(T1,T2). All processes are reset independently between rounds. Let E be the event that process 1 finishes first. You may use…