Homework answers / question archive / 1 Drive-away driving school claims that for each instructor the proportion of students gaining their drivers license at the first-attempt (the pass rate) can be modelled by a continuous random variable, X, with pdf: 3(11—2)* for0<a2<1 fx(x) = | 0 otherwise a Sketch the graph of fx (2) (1) b Find Fx(x) for —co < x < oo and sketch its graph (3) c Find the probability that a randomly chosen instructor has a pass rate of less than a quarter, ie, find P(0 < x# < 0

1 Drive-away driving school claims that for each instructor the proportion of students gaining their drivers license at the first-attempt (the pass rate) can be modelled by a continuous random variable, X, with pdf: 3(11—2)* for0<a2<1 fx(x) = | 0 otherwise a Sketch the graph of fx (2) (1) b Find Fx(x) for —co < x < oo and sketch its graph (3) c Find the probability that a randomly chosen instructor has a pass rate of less than a quarter, ie, find P(0 < x# < 0

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1 Drive-away driving school claims that for each instructor the proportion
of students gaining their drivers license at the first-attempt (the pass
rate) can be modelled by a continuous random variable, X, with pdf:

3(11—2)* for0<a2<1
fx(x) = |
0 otherwise
a Sketch the graph of fx (2) (1)
b Find Fx(x) for —co < x < oo and sketch its graph (3)
c Find the probability that a randomly chosen instructor has a pass
rate of less than a quarter, ie, find P(0 < x# < 0.25). (1)
d Find the probability that at least half the students of a randomly
chosen instructor gaining their drivers license at the first-attempt.
(2)
e Find E(X), the mean pass rate. (2)
f Will the median pass rate be smaller than, greater than or the
same as the mean pass rate?
Justify your answer without calculating the median. (1)
g Verify your answer to 1f by calculating the median pass rate (2)
h Find Var(X) (2)
2 The time to failure for a particular electric car battery can be modelled
by a random variable X with with pdf:
kxb-te-e@"” for r > 0
X\L) =
Ix(@) ‘ otherwise
where a > 0 and 6 > 0.
(Note that enon” = exp(—ax?))
a Express k in terms of a and £. (2)
b Find the pdf of Y = aX? when at least one of the parameters
a and 6 are not equal to 1. Remember to include the range of
values of y for which the result holds. Hence show that Y has an
Exponential(1) distribution. 

3 Suppose we have two continuous random variables X and Y ,with joint
distribution function:
*(6 x-y) for0<4%<2,2<y<4
fxy(a,y) = 4 8 ee
0 otherwise

a Find the marginal probability density functions for X and Y, i.e.
find fx(x) and fy(y). In each case remember to include the range
of values of the appropriate random variable for which the result
holds. (4)
b Find fx)y(x|y). Remember to include the range of values of x
and y for which the result holds. (2)
c Find fy|x(y|z). Remember to include the range of values of x
and y for which the result holds. (2)
d Are X and Y independent? Justify your answer with reference
to 3a, 3b and 3c. (3)

e Show that
Fx y(2,y) = = (6y+@—10-5(«-+y)) if0<a<2and2<y<4
(2)
f Copy and complete the table below for Fx y(z, y). (3)

Y
Fyy(z,y)|y<2 2<y<4 yo4
we fee. Le 1
g Compute P(X <1,Y < 3). (2)
2g

4 Marie has been taking note of the time she leaves for work and the
length of her morning commute. She decides to model the number
of hours X after 6:30am that she leaves for work with a Uniform(2)
distribution, ie X has pdf:

c forO<a%<2
eX) =
Ix(@) ‘ otherwise
where c is a positive constant.
She notices that, when 0 < x < 2 the length of the commute (in hours),
Y, seems to depend upon the time she leaves for work and decides to
model Y|X with a distribution, with pdf:
6+ 8(2-—y) for t +a<y< ares
fy\x(ylz) = 4 4
0 otherwise
Note that fy)x(y|x) is defined only for 0 < x < 2.
a Determine the value of c. (1)
b Find the joint pdf of X and Y, fx y(x,y). Remember to include
the range of values of x and y for which the result holds. (2)
c Find that probability that Marie left home before 7:00am and her
commute is less than 45 minutes. (2)
d Copy and complete the following to rewrite fx y(z,y) with y as
the leading variable. (1)
3+4(2-—y) for ...<y<...,
fxy (x,y) = max(...,0) <a < min(2,...)
0 otherwise
e Find the marginal pdf of Y, fy(y).. Remember to include the
range of values of y for which the result holds. (4) Hint: Use
your answer to 4d,sketch the region over which to integrate and
note that this means that fy(y) will be in the form :
... for 0.25 <y<a,
... forax<y<Qb,
fyy) = ... forb<y < 2.75,
0 otherwise
f Find the probability that Marie’s morning commute is less than
45 minutes. (2)
3

 If Marie’s commute is less than 45 minutes, find that probability
that she left home before 7:00am (2)
h Write down the probability that Marie arrives on time (or early)
for a 7:30 meeting in terms of X and Y. (Note you are not
being asked to calculate the probability, just to write a probability
statement, ie the probability to be calculated.) (1)
Total: 55 marks.
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