2) X and Y have the following joint probability density function:
n=4, X=2, Y=1, Z=3
Find:
a)Marginal distribution of X and Y.
b)Mean of X and Y.
c)E(XY).
d)Covariance of X and Y and comment on it.
e)Correlation coefficient between X and Y. And comment.
2) X and Y have the following joint probability density function:
{((5y^3)/(96x^2)) 2<x<5, 0<y<4
0 Elsewhere}
Find:
a) Marginal distribution of X and Y
b) Mean of X and Y
c) E(XY)
d) Covariance of X and Y and comment on it.
e) Correlation coefficient between X and Y. And comment.
In: Statistics and Probability
Ex 4.
(a) Prove by induction that ∀n∈N,13+ 23+ 33+···+n3=[(n(n+ 1))/2]2
b) Prove by induction that 2n>2n for every natural number n≥3.
In: Advanced Math
to calculate the integral from (-infinty) to (+ infinity) of [x^2 / (x^4 + 4)[ dx . poles, residues.
In: Math
Calculate the integral from (-infinty) to (+ infinity) of [x^2 / (x^4 + 4)[ dx
In: Advanced Math
Consider the ODE y"+ 4 y'+ 4 y = 5 e^(− 2 x ). (
a) Verify that y 1 ( x) = e − 2 x and y 2 ( x) = xe − 2 x satisfy the corresponding homogeneous equation.
(b) Use the Superposition Principle, with appropriate coefficients, to state the general solution y h ( x ) of the corresponding homogeneous equation.
(c) Verify that y p ( x) = 52 x 2 e − 2 x is a particular solution to the given nonhomogeneous ODE.
(d) Use the Nonhomogeneous Principle to write the general solution y ( x ) to the nonhomogeneous ODE.
(e) Solve the IVP consisting of the nonhomogeneous ODE and the initial conditions y(0) = 1 , y 0 (0) = − 1 .
In: Advanced Math
Information from the financial statements of Ames Fabricators,
Inc., included the following:
| On October 15, 2020, the board of directors of Ensor Materials
Corporation approved a stock option plan for key executives. On
January 1, 2021, 23 million stock options were granted, exercisable
for 23 million shares of Ensor's $1 par common stock. The options
are exercisable between January 1, 2024, and December 31, 2026, at
90% of the quoted market price on January 1, 2021, which was $20.
The fair value of the 23 million options, estimated by an
appropriate option pricing model, is $6 per option. Ensor chooses
the option to recognize forfeitures only when they occur. Ten percent (2.3 million) of the options were forfeited when an executive resigned in 2022. All other options were exercised on July 12, 2025, when the stock’s price jumped unexpectedly to $22 per share. Required: 1. When is Ensor’s stock option measurement date? 2. Determine the compensation expense for the stock option plan in 2021. (Ignore taxes.) 3. Prepare the journal entries to reflect the effect of forfeiture of the stock options on Ensor’s financial statements for 2022 and 2023. 5. Prepare the journal entry to account for the exercise of the options in 2025. |
||||||||
Ames’s net income for the year ended December 31, 2021, is
$900,000. The income tax rate is 25%. Ames paid dividends of $5 per
share on its preferred stock during 2021.
Required:
Compute basic and diluted earnings per share for the year ended December 31, 2021. (Enter your answers in thousands (For example, 100,000 should be entered as 100). Do not round intermediate calculations.)
In: Accounting
6. Suppose the production function is given by Q=1/20*L^1/2*K^1/4;price of labor(w) = 0.50 and price of capital (r) = 4: The market price for the output produced is P= 560: (a) Short-run production:
ii. Write down this firm's LONG-RUN cost minimization problem. [Note: In long-run, nothing is fixed, so the firm can choose both labor and capital optimally to minimize its cost.]
iii. Solving the long-run cost minimization problem, we get the long-run cost function C(Q) = (12)*(5Q)^4/3:
Find out the long-run cost of the firm for producing 30 units of output.
iv. Write down the functional forms of the long-run average cost and marginal cost.
v. Find out the profit maximizing choice of output and labor in the long-run. How much is the profit?
In: Economics
Show by induction that for all n natural numbers 0+1+4+9+16+...+ n^2 = n(n+1)(2n+1)/6.
In: Advanced Math
Let ?=2^(2^?)+1 be a prime that n>1
1. Show that ? ≡ 2(mod5)
2. Prove that 5 is a primitive root modulo ?
In: Advanced Math
Consider Matrix A = ([5, 0, 4],[1, -1, 0],[1, 1, 0]). Note that [5, 0, 4] is row 1. [1, -1, 0] is row 2. [1, 1, 0] is row 3.
a) Find all Eigenvalues and Eigenvectors.
In: Advanced Math