Want to make sure i understand. The order of rolls doesn't matter. They are pairwise independent. However, not mutually independent be because P (R_j,k , R_a,b) is 1/6^3 and P (R_j,k) is 1/6 with district indices j,k,a,b
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...
I am getting confusing about the question, the investment increased 0.4 in 3 years, and in this 3 years increasing by the same rate, that is how I understand the question.
the investment is increasing at the same rate in 3 years, and we are looking for the rate,r. In order to get the value after 3 years which is already increase 40%, I think (1+r)^3 is the rate increase over 4 years. Thus, this would equal to 1+0.4
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...
Suppose $\phi : F\rightarrow F$ is an identity homomorphism, then $\ker(\phi)=\{0\}$ is an ideal of $F$ and hence $F/\{0\}\cong F$ by the first ring isomorphism.
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...