Question

1) Suppose that X and Y are two random variables, which may be dependent and Var(X)=Var(Y). Assume that 0<Var(X+Y)<∞ and 0<Var(X-Y)<∞. Which of the following statements are NOT true? (There may be more than one correct answer)

a. E(XY) = E(X)E(Y)

b. E(X/Y) = E(X)/E(Y)

c. (X+Y) and (X-Y) are correlated

d. (X+Y) and (X-Y) are not correlated.

2) S.D(X ± Y) is equal to, where S.D means standard deviation

a. S.D(X) ± S.D(Y)

b. Var(X) ± Var(Y)

c. Square root of Var(X) ± Var(Y)

d. Unable to tell

Answer #1

Solution:

Consider the following two samples:

Y | X | X+Y | X - Y |

1 | 25 | 25 + 1 = 26 | 25 -1 = 24 |

2 | 26 | 26 + 2 = 28 | 26 - 2 = 24 |

3 | 27 | 27 + 3 = 30 | 27 - 3 = 24 |

Both the samples have variance of 1, so V(X) = V(Y). Further, E(Y) = 2 and E(X) = 26

Now, evaluating further we can answer the given questions.

1) E(XY) = (1*25 + 2*26 + 3*27)/3 = 52.67

E(X)*E(Y) = 26*2 = 52

So, the two aren't exactly same. Also, Cov(X, Y) = E(XY) - E(X)E(Y)

So, if two equal, Cov(X, Y) = 0, which is not necessarily true as X and Y might be dependent

Similarly, E(X/Y) = (25/1 + 26/2 + 27/3)/3 = 47/3 = 15.67

However, E(X)/E(Y) = 26/2 = 13, again the two aren't equal

Finally, (X+Y) changes by 2 but (X - Y) is still constant, explaining that change in (X +Y) does not change (X - Y), and so the two might not be correlated.

Thus, 3 statements are not true: (a), (b) and (c)

2) S.D(X +/- Y) = (V(X +/- Y))^{1/2}

V(X +/- Y) = V(X) + V(Y) - 2*Cov(X, Y)

So, S.D(X +/- Y) = (V(X) + V(Y) - 2*Cov(X, Y))^{1/2}

Thus, no option define it correctly

Correct option is (d) unable to tell

Consider two random variables X and Y such that E(X)=E(Y)=120,
Var(X)=14, Var(Y)=11, Cov(X,Y)=0.
Compute an upper bound to
P(|X−Y|>16)

If X, Y are random variables with E(X) = 2, Var(X) = 3, E(Y) =
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(a) For Z = 3X − 1 find µZ, σZ.
(b) For T = 2X + Y find µT , σT
(c) U = X^3 find approximate values of µU , σU

Let X and Y be jointly distributed random variables with means,
E(X) = 1, E(Y) = 0, variances, Var(X) = 4, Var(Y ) = 5, and
covariance, Cov(X, Y ) = 2.
Let U = 3X-Y +2 and W = 2X + Y . Obtain the following
expectations:
A.) Var(U):
B.) Var(W):
C. Cov(U,W):
ans should be 29, 29, 21 but I need help showing how to
solve.

2. Random variables X and Y have a joint PDF fX,Y (x, y) = 2 for
0 ≤ y ≤ x ≤ 1. Determine (a) E[X] and Var[X]. (b) E[Y ] and Var[Y
]. (c) Cov(X, Y ). (d) E[X + Y ]. (e) Var[X + Y ].

Suppose that X and Y are two jointly continuous random variables
with joint PDF
??,(?, ?) =
??
??? 0 ≤ ? ≤ 1 ??? 0 ≤ ? ≤ √?
0
??ℎ??????
Compute and plot ??(?) and ??(?)
Are X and Y independent?
Compute and plot ??(?) and ???(?)
Compute E(X), Var(X), E(Y), Var(Y), Cov(X,Y), and
Cor.(X,Y)

Let X, Y be two random variables with a joint pmf
f(x,y)=(x+y)/12 x=1,2 and y=1,2
zero elsewhere
a)Are X and Y discrete or continuous random variables?
b)Construct and joint probability distribution table by writing
these probabilities in a rectangular array, recording each marginal
pmf in the "margins"
c)Determine if X and Y are Independent variables
d)Find P(X>Y)
e)Compute E(X), E(Y), E(X^2) and E(XY)
f)Compute var(X)
g) Compute cov(X,Y)

Suppose X and Y are independent variables with E(X) = E(Y ) = θ,
Var(X) = 2 and Var(Y ) = 4. The two estimators for θ, W1 = 1/2 X +
1/2 Y and W2 = 3/4 X + 1/4 Y .
(1) Are W1 and W2 unbiased? (2) Which estimator is more
efficient (smaller variance)?

Let X and Y be continuous random variables with joint density
function f(x,y) and marginal density functions fX(x) and fY(y)
respectively. Further, the support for both of these marginal
density functions is the interval (0,1).
Which of the following statements is always true? (Note there
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E[X^2Y^3]=(∫0 TO 1 x^2 dx)(∫0 TO 1 y^3dy)
E[X^2Y^3]=∫0 TO 1∫0 TO 1x^2y^3 f(x,y) dy dx
E[Y^3]=∫0 TO 1 y^3 fX(x) dx
E[XY]=(∫0 TO 1 x fX(x)...

Suppose that the joint probability density function of the
random variables X and Y is f(x, y) = 8 >< >: x + cy^2 0 ≤
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(a) Sketch the region of non-zero probability density and show
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(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...

Let X ~ N(1,3) and Y~ N(5,7) be two independent random
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Var(X -Y)
Var(2X - 4Y)

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