Math Answers

Solve the Cauchy Problem,

xu_x+(x+y)u_y=u+1 with u(x,y)=x² on y=0

The joint probability density function of 𝑋 and 𝑌 is given by

𝑓(𝑥, 𝑦) = {

𝑥 + 𝑦

3

0 ≤ 𝑥 ≤ 1, 0 ≤ 𝑦 ≤ 2

0 𝑒𝑙𝑠𝑒𝑤ℎ𝑒𝑟𝑒

Find the two lines of regression.

Find a relation R such that 𝑥+𝑦 2 >1 if A = {0,1, 2} and B ={0, 1, 2, 3}. 2. Find a relation R such that y is a power of x if A = {1, 2, 3} and B = {1, 4, 5, 9}

2. a) Find the derivatives of the following functions with respect to x.

x ^ 3 + y ^ 3 = 3

y = (sin x) ^ tan x

b) Evaluate the 2 ^ (nd) order partial derivatives partial^ 2 u partial x^ 2 and partial^ 2 u partial y^ 2 if u=2x^ 3 +3x^ 2 y+xy^ +y^ .

Let A be a given finite set and P(A) its power set. Let ⊆ be the inclusion relation on the elements of P(A). Draw Hasse diagrams of (P(A), ⊆) for A={a}; A={a,b}; A={a,b,c} and A={a,b.c.d}.

Let 𝐼 = 0∫10 𝑓(𝑥)𝑑x . You are asked to approximate the value of I using (i) uniform random 0 10 ∫ 𝑓(𝑥)𝑑𝑥 variable, (ii) Monte Carlo simulation and (iii) antithetic variates

a. Write down the procedures for the Monte Carlo simulation without using antithetic variates.

b. Write down the antithetic variable.

A certain firm uses a large fleet of delivery vehicles. Their records over a long period of

time (during which their fleet size utilization may be assumed to have remained suitably

constant) show that the average number of vehicles per day is 3. Estimate the probability

on a given day when

i. all their vehicles will be serviceable (2 Marks)

ii. more than 2 vehicles will be unserviceable (2 Marks)

iii. exactly 4 vehicles will be unserviceable

3.  Jim works in a concession stand three nights a week for 22 weeks. He is allowed to have one free drink, the second drink at half price, and the third at full price. Jim averages three 50-cent drinks each time he works, but does not pay for any. How much money is he not paying per week?  (a) __________ For four weeks?  (b) __________ For 22 weeks?  (c) ________

A hospital employs 250 nurses. In a random sample of 81 of these, the mean number of

hours overtime billed in a particular week was 12.3, and the sample standard deviation was

8.4 hours. Find a 98% confidence interval for the mean number of hours overtime billed

per nurse in this hospital that week?

A simple random sample of 80 items resulted in a sample of 49. The population standard

deviation is σ = 7 .

i. What is the standard error of the mean? (1 Mark)

ii. At 99% probability, what is the margin of error?