Download Mid-term

ENM 500
1.
Midterm Review of Probability
In 5-card poker the probability of 4 aces is 1.847e-5
P(2 pair) is 0.04754
P(3 of a rank) is 0.02113
.
.
2.
Find n, p and q given binomial RV X with  = 5 and 2 = 4.
3.
Five people get on an elevator at the bottom floor in a building with 6 floors.
The probability that all 5 people get off on separate floors is (/ (perm 5 5)
(expt 5 5))
4.
P(4 people sharing a birth month) = 1 – (12*11*10*9)/124.
6.
In a box are 5 brown, 6 green and 9 red socks. You grab 2 in the dark.
P(matching pair) is
.
(/ (+ (comb 5 2) (comb 6 2) (comb 9 2))
(comb 20 2))
7.
ASK NOT WHAT YOUR COUNTRY CAN DO FOR YOU; ASK WHAT YOU CAN
DO FOR YOUR COUNTRY. RV X is the number of letters in a word.
V(X) =
.
X
P(x)
1
2
3
4
5
6
7
8.
P(2 S6) in 5 tosses of a pair of fair dice = (binomial 5 5/36 2).
9.
A dresser has 3 drawers. In one drawer is a 2-headed coin, in another is a 2tailed coin, and in the third drawer is a fair coin.
P(faces match) =
.
You randomly choose a drawer and open it to see that the face showing is
heads. P(Hidden face is heads) =
.
1
10.
RV X has density f(x) = 2x on [0, 1], and Y = 3X - 1.
Then Y’s density is
. E(Y) = E(3x – 1) = 3 * 2/3 – 1 = 1.
f(y) = 2(y + 1)/9 on [-1, 2]
2 2 2
2 y 3 y 2 2 2 8 4 1 1
E(Y) =  ( y  y )dy  (  ) |1  (  
 ) 1
9 1
9 3 2
9 3 2 3 2
11.
Suppose earthquake occurrences average 2 per year in California.
P(2 or more occurrences in next 2 years is (- 1 (cpoisson 4 1)).
12.
RV X has density function f(x) = 3e-3x for x > 0. Find the 70 percentile.

x
0
3e3 x dx  0.7 Solve for x.
13.
Compute the mean, median, mode, IQR, standard deviation, and variance
from the following set of data: 2 3 7 5 8 4 -8 12 0 2.
14.
Find the mean, median, and mode of RV X distributed f(x) = 2x on [0, 1].
15.
Given the joint density below, find fY|X=0,Z=1.
X
0
1
1
0
Y
1
1
0
0
Z
1
0
0
1
fXYZ
½
1/8
¼
1/8
16.
X
y/x
y/x = 0, z=1
0
3/8
1/5
1
5/8
4/5
G(X) = X2 + 3X + 2. Find E[G(X)] if
a. RV X is N( = 3, 2 = 4)
b. RV X is Poisson  = 3.
17.
V(X) = 2 and V(Y) = 5 and C(X, Y) = -2. Find V(2X - 3Y).
18.
Given joint density fXY (x,y) = 6xy2 for 0 ≤ x ¸1; 0 ≤ y ≤ 1, find E(Y|X).
19.
Y = 3X - 5 where RV X has density f(x) = 1.5 x on [0, 1]. Find E(Y) and fY.
2
20.
RV R, the radius of a circle, is exponential. Find fA for RV A, the area.
ENM 500
Review Midterm 2010
1. Randomly cut any length wire and determine probability that longer piece is
twice shorter piece. P(longer piece > ½) = _________
2. Probability of throwing a pair of fair dice and getting a sum of 7 or a 5 before
an 8 or a 9.
3. Write f(x) given Joint density f(x, y) = x + y; x, y on [0, 1].
Find P(X < ½, Y < ½) and fx(x)
f(x) =
1
 ( x  y)dy  x  1/ 2 on [0, 1]
0
P(X < ½, Y < ½) =
1/2
1/2
0
0
 
1/2 x
1
1
( x  y )dydx   (  )dx 
0
2 8
32
4. Poker, n-gons and shoes
P(4 aces and a king in 5-card poker) = 4/2598960
The number of diagonals of a 10-gon is _____.
P(2 pair from 5 styles in selecting 6 shoes [xx yy z w]) =
5 * 6 * 2 * 2 / 210
5. RV Y = 5X – 6 for f(x) = 3x2 on [0, 1]. Find E(Y) and f(y).
E(Y) = -9/24
Y's density function is f(y) = 3(y + 6)/25 on [-6 -1]
3
6.
How many ways can 7 marbles be put into 3 bins?
(comb 9 2)  36 ways
.
List them. 007 016 025 034 115 124 133 223
3 + 6 + 6 + 6 + 3 + 6 + 3 + 3 = 36
How many ways with at least one in each bin?
Take 9 (7 + 3 - 1) items and put one in each of 3 bins
leaving 6 items to distribute as (comb 6 2) = (comb 6 4)  15
115 124 133 223
3 + 6 + 3 + 3 = 15
7. Given independent RVs X ~ N(4, 16) and Y ~N(40, 9)
a) P(X > 4, Y  40) =
.
b) P(X > 4 OR Y < 40) =
.
c) P(X < 2, Y > 43) =
.
d) P(Y - X < 38 =
.
e) The value of X at the 90th percentile is _____.
9.
How could one get a random sample from the density function
f(x) = 2x2 on [0, 1.51/3)?
1/3
 3Y 
Y = F(X) = 2X /3 => X = 

 2 
3
E(X) = 0.85853568 (theoretical)
(mu (mapcar #' cube-root (mapcar #' * (list-of 100 3/2)
(sim-uniform 0 1 100))))
 0.860257, 0.847954 …
4
10.
Find the number of randomly selected people to ask so as to have a
probability of ¾ that at least one person in the group shares your birthday.
P(not sharing) = 364/365
P(n not sharing) = (364/365)n = 1 - ¾ = ¼.
n(-2.743486) = -1.38629 => n  506 people
11. If the variance of the population {a b c} is 12, then
the variance of population {2a 2b 2c} is ________.
12. (phi 1 )  ______
(Del-phi -1 1)  ______
13. How many diagonals does a 10-gon have?
14. P(S9 occurs before S8) in throwing a fair pair of dice.
15. How many ways can you put 5 marbles into 3 bins?
16. You randomly select 4 numbers from the numbers 1 to 20.
Compute probability that 10 is the 2nd smallest number.
17. Arrivals are a Poisson RV X process with 10 per hour.
Find P(X = 10) in next hour. (poisson 10 10)
Find probability of no arrivals in next hour. (poisson 10 0)
Verify using the Exponential. (U-exponential 10 1)
18. How many unique 5-digit integers are there with no leading zero and sampling
without replacement? How many are even?
-
19.
9 * 9 * 8 * 7 * 6 Total = 27216
8 * 8 * 7 * 6 * 5 Odd = 13440
Even = 13776
Given joint density fXY (x,y) = 6xy2 for 0 ≤ x ¸1; 0 ≤ y ≤ 1, find E(Y|X). = 3/4
5
20.
Y = 3X - 5 where RV X has density f(x) = 1.5
x on [0, 1]. Find E(Y) and fY.
E(Y) = -16/5 fy = ½*[sqrt (y+5/3)].
21.
RV R, the radius of a circle, is exponential. Find fA for RV A, the area.
A = r2; dA = 2r dr; dr/dA = 1/2r
f(r) = ke-kr r > 0
fA = ke-k(sqrt A/) (1/2r)
= k(sqrt a/)e-k(sqrt a/)
22.
Find the probability by writing the canonical patterns for throwing 4 fair dice
and getting a sum of 6.
1113 1122 => 4!/3! + 4!/2!2! = 4 + 6 = 10; 10/1296
23.
Given RV X distributed f(x) = c on [3, 5], find:
a) c
b) E(X)
c) V(X)
e) Median(X)
f) 78th percentile
d) P(3 < X < 4)
g) the x for which 80% exceed it.
24.
Given joint density f(x, y) = 2x for 0 < x < 1; 0 < y < 1, find E(Y); V(Y);
E(X), V(X), and .
25.
Explain using Lisp code how to simulate a value for the cos xdx
26.
Four playing cards, 2 red and 2 black are lying face down on the table. You
randomly choose any two cards. P(Match) = ____

a)
b)
c)
d)
Either match or don't match, so 50/50
Both red, both black, one of each, 2 out of 3 match, so 2/3
BB BR RB RR: so 1/2
None of the above (permute '(R1 R2 B1 B2))  8/24
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