Homework Statement
Roll a fair die 5 times, find the probability that the first two rolls have the same outcomes.
Homework Equations
The Attempt at a Solution
The total outcomes is 6^5, I think we have 6^2 * 6 choose 2 /6^5 since the first two numbers are fixed and we can choose 2 numbers...
Homework Statement
Let Y_1,Y_2 be independent random variable with uniform distribution on the interval [1,2]. Define X=max{Y_1,Y_2}. Find p.d.f., expected value and variance.
Homework Equations
The Attempt at a Solution
Since $X=\max\{Y_1,Y_2\}$, this tells $Y_1$ and $Y_2$ must at most $x$...
Homework Statement
It rains in a city with a chance of 0.4. The weather forecast is not always accurate. When there will be a rain the next day, the forecast predicts the rain with probability 0.8; When there is no rain, the forecast falsely predicts a rain with probability 0.1. You take your...
Hi, all. I am planning to apply REU for math this summer. This is my first time to write a personal statement, I am not sure how it is. Basically, my statement needs to answer why I want to apply this program and what my future goal is.
Malcolm X said: “Education is our passport to the future...
Homework Statement
Let F = {0,1,α,α+1}. Find all irreducible polynomials over F of degree at most 2.
Homework Equations
The Attempt at a Solution
To determine an irreducible polynomial over F, I think it is sufficient to check the polynomial whether has a root(s) in F,
So far, I got...
Homework Statement
If the function f:ℝ→ℝ is differentiable and f(x)<=f(0) for all x ∈[-1,1], then f'(0)=0. True or False.
Homework Equations
The Attempt at a Solution
I think the statement is right. Since f(x)<= f(0) for all x in [-1,1], this tells us f is an even function or a symmetric...
Homework Statement
A one to one function f: ℝ→ℝ is monotone, True or False
Homework Equations
The Attempt at a Solution
I think the statement is false, for example: Let I =[0,1]∪[2,3] f(x)=x if x∈[0,1], f(x)=5-x,x∈[2,3]
Homework Statement
The state of Oregon wishes to design a new lottery game with the following rules: 1.Each ticket costs $5 2.There will be three prizes: $10, $100 and $1000 3.The probability of the $10 prize will be 20%. 4.The probability of the $100 prize will be 1% 5.Ten thousand...
Homework Statement
If an investment increased by 40% over 3 years and rose by the same percentage rate each of those years, what was the rate?
Homework Equations
new_value = old_value(1+r)^p
The Attempt at a Solution
Let value of the investment be 1000, then we have 1000(1+0.4) = 1000(1+r)^3...
Homework Statement
a. Lucy's Lunch and Latte has found that customers are put off by the local tourism tax of 9% that is added to their bill. If Lucy decides to cover the tax herself, rather than adding it to the customer's bill, what percent will the customer see in savings?
b. Lucy decides...
Homework Statement
S≡{x|x∈ℝ,x≥0,x2 < c} Prove Sup S = c
Homework Equations
The Attempt at a Solution
Since x in the set real numbers, there are two cases for x: x < 1 or 0 <= x <=1
if 0 <= x <=1, then x < c + 1 since c is positive.
if x < 1, then x^2 < c < x*c + x = x(c+1)
thus x < c+1, by...
Homework Statement
Let F be a finite field of characteristic p∈{2,3,5}. Consider the quaternionic ring, Q_F={a_1+a_i i+a_j j+a_k k|a_1,a_i,a_j,a_k ∈ F}. Prove that Q_F is not a division ring.
Homework Equations
The Attempt at a Solution
Let α=1+i,β=1+i+j∈QF. Then...
Homework Statement
Show √ 2 + √ 3 algebraic over Q. Find its degree over Q. Prove the answer.
Homework Equations
The Attempt at a Solution
Let ##\alpha= \sqrt{2}+\sqrt{3}\in \mathbb{R}##, then ##\alpha^4-10\alpha^2+1=0## which is a root of ##f(x)=x^4-10x^2+1## where ##f(x)## in...
Homework Statement
Construct $\mathbb{F}_{16}$ as a quotient of $\mathbb{Z}_2[X]$. How many non-zero elements are primitive in this field? Calculate $|GL2_(\mathbb{F}_16)|$.
Homework Equations
Primitive Theorem
The Attempt at a Solution
For the first question, I don't know how to construct...
Homework Statement
Let F be a field. Show that F is isomorphic to F/{0}
Homework Equations
The Attempt at a Solution
By the first ring isomorphic theorem, kernel of the homomorphism is an ideal which is either {0} or I. Hence F isomorphic to F/{0}
I think I misunderstood the problem can...
Homework Statement
Show that the group of units in Z_10 is a cyclic group of order 4
Homework Equations
The Attempt at a Solution
group of units in Z_10 = {1,3,7,9}
1 generates Z_4
3^0=1, 3^1=3, 3^2=9, 3^3= 7, 3^4= 1, this shows <3> isomorphic with Z_4
7^0=1 7^1= 7, 7^2= 9 7^3=3 7^4=1...
Homework Statement
Let G be any group and a in G, define f: Z → G by f(n) = a^n
Apply any isomorphism theorem to show that range of f is isomorphic to a quotient group of Z
Homework Equations
The Attempt at a Solution
The range of f is a^n , then quotient group of Z is Z/nZ
Apply the...
Homework Statement
Question: There is a fixed 55 cm long rod (vertical) that is charged uniformly up to +137 nC, Place a 0.5 g plastic 6 cm long stirrer (horizontal) that is charge to +93 nC, 2.3 cm away from the rod
Calculate the Electric force and acceleration on the stirrer .
Homework...
Homework Statement
Charge a ring of radius R=5.0cm laying in the x-y plane to 50nC.
Create a VPython program that will allow you to calculate the E-field due to the ring anywhere in space.
Homework Equations
E_ring=kQz/(R^2+z^2)^(3/2), point P above the xy-plane
The Attempt at a Solution...
Homework Statement
Let, M={ (a -b) (b a):a,b∈ℝ}, show (H,+) is isomorphic as a binary structure to (C,+)
Homework Equations
Isomorphism, Group Theory, Binary Operation
The Attempt at a Solution
Let a,b,c,d∈ℝ
Define f : M→ℂ by f( (a -b) (b a) ) = a+bi
1-1:
Suppose f( (a -b) (b a) )= f( (c...
Homework Statement
Taking light collected by the telescope and shine it through a diffraction grating with a 765.5 lines/mm. Using the filter that just the brightest Helium II line. you cast the diffraction grating's diffracted pattern on a screen is 22.000cm away and there are 3 dots, m=-1...
Homework Statement
What is the magnitude of the net gravitational force on the m2=5kg mass? Assume m1=25kg and m3=15kg.
https://session.masteringphysics.com/problemAsset/1383524/6/13_P28.png
Homework Equations
F=GMm/r^2
The Attempt at a Solution
First I find the angle between m1 and m2 which...
Homework Statement
You've taken a summer job at a water park. In one stunt, a water skier is going to glide up the 2.0-m-high frictionless ramp shown, then sail over a 5.0-m-wide tank filled with hungry sharks. You will be driving the boat that pulls her to the ramp. She'll drop the tow rope at...
Homework Statement
let A be the matrix corresponding to the linear transformation from R^3 to R^3 that is rotation of 90 degrees about the x-axis
Homework Equations
find the matrix A
The Attempt at a Solution
I got stuck on rotating z component.
I tried T([e1,e2,e3])=[0 -1 0]...
Homework Statement
A ball is thrown toward a cliff of height h with a speed of 28.0 m/s and an angle of 60.0° above horizontal.
What is the ball's impact speed
Homework Equations
Vf=Vi-at
Sf=Si+Vit-1/2gt^2
Vy=sintheta
Vx=costheta
The Attempt at a Solution
first find the vertical velocity =...
Homework Statement
the minimum distance for a car traveling at a speed of 30m/s is 60 m, including the distance traveled during the driver's reaction time of 0.44s. what is the minimum stopping distance for the same car travelling at the speed of 41m/s?
v=30m/s, s=60m
drivers' reaction time =...