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Name:
Math (^) 29— (^) Probability
Practice
(^) Final (^) Exam
Instructions:
Show (^) all (^) work.
(^) You (^) may (^) receive partial
(^) credit
(^) for (^) partially
(^) completed
(^) problems.
You may^ (^) use (^) calculators
(^) and (^) a two-sided
(^) sheet
(^) of (^) reference
(^) notes.
(^) You (^) may (^) not (^) use (^) any (^) other
references
(^) or (^) any (^) texts, (^) except
(^) the (^) provided
(^) z-table.
You (^) may (^) not (^) discuss
(^) the (^) exam (^) with (^) anyone
(^) but (^) me.
Suggestion:
(^) Read (^) all (^) questions
(^) before
(^) beginning and
(^) complete
(^) the (^) ones (^) you know
(^) best (^) first.
Point (^) values
(^) per (^) problem
(^) are (^) displayed
(^) below
(^) if that (^) helps (^) you (^) allocate
(^) your (^) time among
problems.
You (^) need (^) to (^) demonstrate
(^) that (^) you (^) can (^) solve (^) all (^) integrals
(^) in (^) problems
(^) that (^) do (^) not (^) have (^) a (DO
NOT
SOLVE)
(^) statement.
I.E.
(^) write
out^ (^) some (^) work (^) showing
(^) how (^) you (^) solved
(^) the (^) integration,
including
(^) if necessary
(^) integration
(^) by (^) parts.
Good (^) luck!
Problem
Total
Points (^) Earned
Possible
(^) Points
Note: (^) The (^) points
(^) total (^) on (^) your (^) final (^) will (^) be (^100) (^) points.
(^) This (^) was (^) my (^) fast (^) attempt
(^) to (^) assign
(^) points here,
(^) so
you (^) can (^) see
what^
(^) I (^) thought
(^) the (^) problems
(^) were (^) worth
(^) before
(^) any (^) adjustments.
3
‘
Three (^) bowls
(^) are (^) labeled
(^) 1, (^) 2, (^) and (^) 3, (^) respectively.
(^) Bowl (^) i contains
(^) i (^) white
(^) and (^) 5i (^) red (^) balls, (^) In (^) an
experiment,
(^) a bowl (^) is (^) randomly
(^) selected
(^) from (^) the (^) set (^) of (^) three (^) bowls.
(^) Then,
(^3) balls (^) are (^) randomly
selected without
(^) replacement
(^) from (^) the (^) contents
(^) of (^) the (^) selected
(^) bowl.
a. Given (^) that (^) bowl (^1) was (^) NOT (^) selected,
(^) what (^) is the (^) probability
(^) of (^) drawing
(^) exactly
(^2) red (^) balls?
I ()
(2) ___
il
—
()
2(c)
o
b. What (^) is (^) the (^) probability
(^) that (^) exactly
(^2) red (^) balls (^) are (^) drawn?
ci
)
(
)
()
L J)
/
‘,.
—
3c)
—
3t
3
c. Given (^) that (^) exactly
(^2) red (^) balls (^) were (^) drawn,
(^) what (^) is (^) the (^) probability
(^) that (^) bowl
3 was (^) selected?
1
P
(2t)
P(
I
1
o)
P(L
,1I
i
3 )(V 3 )
y 3
(2)
l
X
amrna
(^) (
Consider
(^) a random
(^) variable
X
with (^) pdf (^) given (^) byf(x)
(^) =
kxe
(^) “i, x>O,^ (^) and (^) 0, (^) otherwise.
a. Find (^) the (^) value
(^) of (^) k that (^) makes
(^) this (^) a valid pdf.
ynuS
bL
b. Find (^) the (^) mgf (^) for (^) X. (^) You (^) may (^) either
(^) identify
(^) the (^) distribution
(^) and provide
(^) its (^) associated
(^) mgf (^) or (^) derive
the (^) mgf (^) directly.
a
S
(l- 1 3t)
( 7 IH
c. Find (^) the (^) mean
(^) and (^) variance
(^) of (^) X using (^) the (^) moment
(^) generating
(^) function
(^) you (^) found
(^) in (^) b.
= 2(Lf)
ç
= ‘ (^) (i-’i)
=
(Xz)
(x)-
Lx)]
32
(^2)
32
V
A
company needs
(^) a vast (^) amount
(^) of (^) iron (^) ore (^) for (^) a project.
(^) Suppose
X 1
,X 2 ,...,X 40
(^) are (^) a random
sample
(^) of (^) measurements
(^) on (^) the (^) proportion
(^) of (^) impurities
(^) in (^) iron (^) ore samples from
(^) “Ores
(^) R Us” (^) (a
supplier
(^) company).
(^) The (^) proportion
(^) of (^) impurities
(^) in the^ (^) population
(^) of (^) all (^) similar
(^) iron (^) ore (^) samples,
X,
(^) has
pdf
f(x)^
(^) =
3x 2 ,O
(^) <x (^) <I, (^) and (^) 0, (^) otherwise.
,
(
a. The (^) company
(^) will (^) refuse
(^) to (^) buy (^) the (^) ore (^) if X (^) exceeds
(^) Find (^) the (^) approximate
(^) probability
that^
Xexceeds.8forasampleofsize40.
=
CLI>
N(
/v)
o3O
__
=
b. What (^) numerical value
(^) does (^) X 40 (^) converge
(^) in (^) probability
(^) to? (^) Justify
(^) your answer.
v(x)
b. Show (^) that (^) X and (^) Y are (^) independent
(^) and (^) have (^) identical
(^) distributions
(^) (provide
(^) the (^) marginal pdf
(^) they
io
Lj
Y
c. The (^) two (^) counties
(^) want (^) to (^) hire (^) a single (^) company
(^) for (^) the (^) repairs.
(^) One (^) particular
(^) company
(^) will (^) only
handle combined jobs
(^) of (^) at most
(^6) miles (^) at (^) a time (^) for (^) a given (^) week (^) before
(^) charging
(^) huge (^) additional
(^) fees.
Using (^) a probabilistic
(^) argument
(^) (i.e. (^) compute
(^) a meaningful probability),
(^) would
(^) you (^) recommend
(^) the
counties
(^) use (^) this (^) company
(^) for (^) their (^) repairs?
share).
(j
fx
‘
/
\
9
x
,‘i Cl
2\I
x
0
1 —
)O
xY
-(
‘I
-C
//
11 zI
Jo
Ifo
( (^) -
12X
z)
L
‘I
2 9(;)
15
Suppose
X
and (^) Y are random
(^) variables
(^) where
(^) Var(X)=
(^) and (^) Var(Y)=6.
a. If (^) X (^) and (^) Y are (^) independent,
(^) what (^) is (^) the (^) variance
(^) of6X
(^) + 3Y (^) + 2?
v(xV(Y)
397
b. If (^) X (^) and (^) Y have (^) correlation
(^) what (^) is (^) the (^) variance
(^) of
X^
(^) — 2Y?
Coy (^) (X)
.
V’X2Y)
H/)_(21])
20,
c. (A Little (^) Theory)
(^) If (^) X and (^) Y are (^) independent
(^) random
(^) variables,
(^) show (^) that
E(Y 3
I
X)
(^) =
E(Y 3 ).
(^) You
may (^) treat (^) X and (^) Y (^) as (^) continuous
(^) random
(^) variables, and
use^ (^) regular
(^) notation
(^) for (^) their (^) joint (^) pdf and
marginal
(^) pdfs.
I
E(YIX)
j
3
j
(7y)
Ico
X
b’c
Y (^) ±
The (^) class (^) is (^) throwing
(^) a celebratory
(^) party for
(^) the (^) end (^) of (^) the (^) semester.
A
(^) large (^) number
(^) of (^) pizzas
(^) are
ordered
(^) —40%
(^) from (^) Antonio’s
(^) and (^) the (^) rest (^) from (^) Domino’s.
(^) Of (^) the (^) Domino’s
(^) pizzas,
(^) 30% (^) are (^) cheese
(^) only
while (^) the (^) rest (^) have (^) some
toppings.^
(^) From (^) Antonio’s,
(^) only (^) 15% (^) are (^) cheese
(^) only. (^) What
(^) is (^) the (^) probability
that (^) a pizza (^) came
(^) from (^) Antonio’s
(^) if it (^) is (^) known
(^) to (^) have (^) toppings besides
(^) cheese?
p(o
p(ciD)=
,
p(cA)
J
p(A_d
p(A)P(’I\C1A)
p(c)
(A)
?(R,4)
(D)
p(Nci
—
Matching.
(^) (Not (^) all (^) choices
may^ (^) be (^) used.)
1), A stochastic
(^) process
(^) where the
(^) random
(^) variables
A.
Normal
are (^) related
(^) by (^) conditional
probabilities^
B.
Combination
C.
Permutation
A. (^) A (^) distribution
(^) that (^) may (^) be (^) used (^) to approximate
(^) the (^) Poisson
Markov
(^) Chain
E.
Poisson
Result (^) related
(^) to convergence
(^) in (^) probability
F.
Cauchy
G.
Weak (^) LLN
C. Combinatorial
(^) method
(^) that (^) is (^) employed when
H.
Gamma
order (^) of (^) objects
(^) in (^) a subset
(^) does (^) matter
SampleI.^
(^) Mean
F Example
(^) distribution
(^) where
(^) the mean^ (^) doesn’t
(^) exist
E. Example
(^) distribution
(^) where
(^) the (^) mean
(^) and (^) variance
are (^) equal
