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Continuous random variables and the normal distribution — Question 329

Original QCE Vault practice · 7 marks

Q329 · Practice questionTechnology-freeComplex unfamiliar7 marks

QUESTION 329 (7 marks)

A visitor's arrival time XX hours has density f(x)=34x(2−x)f(x)=\dfrac34x(2-x) for 0≤x≤20\leq x\leq2, and zero otherwise. An organiser chooses a half-hour prize window [a,a+1/2][a,a+1/2], where 0≤a≤3/20\leq a\leq3/2.
Density f(x)=3x(2-x)/4 on [0,2]. A half-hour window [a,a+1/2] is hatched; a is not numerically specified.
a)
Write the probability P(a)P(a) of arrival during the prize window as a definite integral.
[1 mark]
b)
Determine the value of aa that maximises P(a)P(a), with justification.
[4 marks]
c)
Determine the exact maximum probability.
[2 marks]
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