Question

2.X, defined over the range (1, ∞) has pdf given by:

f(x) = 2/x^{3}

- Calculate E(x), E(x
^{2}) and Var(x) - Calculate E(X/5) E[(X/5)
^{2}] and Var(X/5)

Answer #1

suppose that x is a continuos randoms variable with pdf given by
f(x) = x3 / 4 for 0 less than or equal to x less than or
equal to 2. Consider random variables defined by function Y= square
root of x
1) what is the support of y?
2) graph y as a function of x
3) give pdf of y
4) what is expected value of y

Let X =( X1,
X2, X3 ) have the joint pdf
f(x1, x2,
x3)=60x1x22, where
x1 + x2 + x3=1 and
xi >0 for i = 1,2,3. find the
distribution of X1 ? Find
E(X1).

Consider the joint pdf
f(x, y) = 3(x^2+ y)/11
for 0 ≤ x ≤ 2, 0 ≤ y ≤ 1.
(a) Calculate E(X), E(Y ),
E(X^2), E(Y^2), E(XY
), Var(X), Var(Y ),
Cov(X, Y ).
(b) Find the best linear predictor of Y given
X.
(c) Plot the CEF and BLP as a function of X.

A random variable X has probability density function f(x)
defined by f(x) = cx−6 if x > 1, and f(x) = 0, otherwise.
a. Find the constant c.
b. Calculate E(X) and Var(X).
c. Now assume Z1, Z2, Z3, Z4 are independent RVs whose
distribution is identical to that of X. Compute E[(Z1 +Z2 +Z3
+Z4)/4] and Var[(Z1 +Z2 +Z3 +Z4)/4].
d. Let Y = 1/X, using the formula to find the pdf of Y.

if X1, X2 have the joint pdf
f(x1, x2) = 4x1(1-x2) ,
0<x1<1
0<x2<1
and 0,
otherwise
1- Find the probability
P(0<X1<1/3 , 0<X2<1/3)
2- For the same joint pdf, calculate
E(X1X2) and E(X1 +
X2)
3- Calculate
Var(X1X2)

The polynomial f(x) given below has 1 as a
zero.
f(x)=x3−3x2+4x−2?

Which functions fit the description?
function 1: f(x)=x^2 + 12. function 2: f(x)= −e^x^2 - 1.
function 3: f(x)= e^3x function 4: f(x)=x^5 -2x^3 -1
a. this function defined over all realnumbers has 3 inflection
points
b. this function has no global minimum on the interval (0,1)
c. this function defined over all real numbers has a global min
but no global max
d. this function defined over all real numbers is non-decreasing
everywhere
e. this function (defined over all...

Suppose the pdf of a random variable X is defined as:
f(x) = (x/16) + (1/4)
for -4 < x <= 0, and
f(x) = -((x^2)/36) + (1/4)
Find the cdf of X.

A geometric distribution has a pdf given by P(X = x) = p(1-p)^x,
where x = 0, 1, 2,..., and 0 < p < 1. This form of the
geometric starts at x=0, not at x=1. Given are the following
properties: E(X) = (1-p)/p and Var(X) = (1-p)/p^2 A random sample
of size n is drawn, the data x1, x2, ..., xn.
Likelihood is p = 1/(1+ x̄))
MLE is p̂ = 1/(1 + x̄))
asymptotic distribution is p̂ ~...

f'(–2
) and (ii)
f"(–2 ) , where
f(x) = √ 5 –
x2 – x3

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