Question

Two entities, X and Y, were identified as sole suspects for a cryptocurrency hack that pillaged $3 billion. Preliminary investigation determined a probability of 0.76 that X is guilty, and 0.24 probability that Y is responsible. After analyzing Tor traffic, investigators found that the perpetrator used a server, Spectre, which is accessible to 80 percent of Tor users. Suppose that Y was discovered possessing access to Spectre. There is no information regarding X's ability to access Spectre.

a). Given Y 's ability to access Spectre, what is the probability that Y carried out the hack? [Hint: Suppose that B = Y hacked, A = access to Spectre, You want P(B|A) = P(A|B) P(B)/(P(A|B)P(B) + P(A|~B)P(~B)) ]

b). Conditional on this information, what is the probability of X possessing access to Spectre? [Hint: Suppose that E = X has access to Spectre, and H = X hacked conditional on info from part (a) To be clear, P(H) = 1 - your answer for part (a) You want P(E) = P(E|H)P(H) + P(E|~H)P(~H) ]

Answer #1

**Answer:-**

**Given
That:-**

Two entities, X and Y, were identified as sole suspects for a cryptocurrency hack that pillaged $3 billion. Preliminary investigation determined a probability of 0.76 that X is guilty, and 0.24 probability that Y is responsible. After analyzing Tor traffic, investigators found that the perpetrator used a server, Spectre, which is accessible to 80 percent of Tor users. Suppose that Y was discovered possessing access to Spectre. There is no information regarding X's ability to access Spectre.

P(Elliptic_Cracker being guilty) = 0.76

P(Bit_Basher being guilty) = 0.24

P(Using Spectre) = 0.80

a) Given that Bit_Basher accessed spectre

Thus P(Bit_Basher carried out the hack) =

b) Given the same information

Thus P(Elliptic_Cracker carried out the hack) =

The following is a 3 x 3 two-way table:
X = 1
X = 2
X = 3
Total
Y = 1
A
B
C
D
Y = 2
E
F
G
H
Y = 3
I
J
K
L
Total
M
N
O
P
According to this table:
a)
A
P
is a joint or conditional or marginal probability.
b)
N
P
is a joint or conditional or marginal probability.
c)
F
H
is a joint or conditional or...

5.1.8 Determine the value of c that makes the function f(x, y) =
c(x + y) a joint probability density function over the range 0 <
x < 3 and x < y < x + 2. c = (give the exact answer in the
form of fraction) Determine the following. Round your answers in
a-f to four decimal places.
a. P(X < 1, Y < 2) =
b. P(1 < X < 2) =
c. P(Y > 1) =...

Suppose X and Y are independent Geometric random variables, with
E(X)=4 and E(Y)=3/2.
a. Find the probability that X and Y are equal,
i.e., find P(X=Y).
b. Find the probability that X is strictly
larger than Y, i.e., find P(X>Y). [Hint: Unlike Problem 1b, we
do not have symmetry between X and Y here, so you must
calculate.]

Suppose that the joint probability density function of the
random variables X and Y is f(x, y) = 8 >< >: x + cy^2 0 ≤
x ≤ 1, 0 ≤ y ≤ 1 0 otherwise.
(a) Sketch the region of non-zero probability density and show
that c = 3/ 2 .
(b) Find P(X + Y < 1), P(X + Y = 1) and P(X + Y > 1).
(c) Compute the marginal density function of X and Y...

Suppose the continuous random variables X and Y have joint pdf:
fXY (x, y) = （1/2）xy for 0 < x < 2 and x < y < 2 (a)
Find P(X < 1, Y < 1). (b) Use the joint pdf to find P(Y >
1). Be careful setting up your limits of integration. (c) Find the
marginal pdf of Y , fY (y). Be sure to state the support. (d) Use
the marginal pdf of Y to find P(Y...

The joint probability distribution of two random variables X and
Y is given in the following table
X Y →
↓
0
1
2
3
f(x)
2
1/12
1/12
1/12
1/12
3
1/12
1/6
1/12
0
4
1/12
1/12
0
1/6
f(y)
a) Find the marginal density of X and the marginal density of Y.
(add them to the above table)
b) Are X and Y independent?
c) Compute the P{Y>1| X>2}
d) Compute the expected value of X.
e)...

Suppose that P{X=i,Y=j} = c(i+j) for non negative integers i and
j with i+j <= 3; otherwise, the probability is zero.
(a) Determine c.
(b) Compute the marginal p.m.f. of X.
(c) Compute the marginal p.m.f. of Y.
(d) Are X and Y be independent?
(e) Compute P(X+Y<2).
(f) ComputeE[XY].
(g) Compute E[X] and E[Y].
(h) Compute E[X+Y].

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