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1. ### Roll a fair die 5 times

Yes, j,k,a,b gives the same outputs, we have 4 rolls have the same outputs.
2. ### Roll a fair die 5 times

j,k,a,b are distinct indices, P (R_j,k and R_a,b) means j,k,a,bth rolls have the same outcomes
3. ### Roll a fair die 5 times

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
4. ### Roll a fair die 5 times

If we let R_j,k be event that jth and kth rolls have the same outcome, then events R_j,k are't pairwise independent.
5. ### Roll a fair die 5 times

I mean _ _ _... I _ _ _..._ j _ _ _..._ _k... something like this.
6. ### Roll a fair die 5 times

Now , suppose we roll a die n times, would the probability that any ith, jth, kth, lth roll have the same outcomes still 1/6?
7. ### Roll a fair die 5 times

I see that already, we have 6/36* 6^3/6^3=1/6

9. ### Roll a fair die 5 times

I can have a combination of any 3 number of 6 if I understand correctly
10. ### Roll a fair die 5 times

I don't understand your second question. The probability of two rolls of a die are the same is 1/6
11. ### Roll a fair die 5 times

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...
12. ### Expected value and variance of max{Y_1,Y_2}

For x from 1 to 2, (x-1)^2 For x below 1, it is 0 For x greater than 2, it is 1
13. ### Expected value and variance of max{Y_1,Y_2}

The cdf is (x-1)^2?
14. ### Expected value and variance of max{Y_1,Y_2}

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 $x$...
15. ### Conditional Probability and law of total probability

I never learn how to draw a tree
16. ### Conditional Probability and law of total probability

I don't see how to draw probability tree.
17. ### Conditional Probability and law of total probability

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...
18. ### Need help for REU personal statement

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...
19. ### Find all irreducible polynomials over F of degree at most 2

α^2=α+1,(α+1)^2=α, the ch(F)=2
20. ### Find all irreducible polynomials over F of degree at most 2

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...
21. ### Does f'(0)=0 where x in [-1,1] and f(x)<=f(0)

At x =0, we still have f'(0)=0?
22. ### Does f'(0)=0 where x in [-1,1] and f(x)<=f(0)

let h(x)=(f(x)-f(0))/x, when x>0, h(x)<=0; when x<0, h(x)=>0; f(x) is bounded above and the supremum is at x = 0, so f'(0)=0
23. ### Does f'(0)=0 where x in [-1,1] and f(x)<=f(0)

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...
24. ### One to one function is monotone?

Let f(x)=x, x is rational, f(x)=-x,x is irrational, the function is one to one,but it is jumping. Does this example apply?
25. ### One to one function is monotone?

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]
26. ### Find the probability of a prize

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...
27. ### Find the percentage rate

Thank you for checking my answer and help me to understand the question in more detail
28. ### Find the percentage rate

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.

I get 1.4
30. ### Find the percentage rate

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
31. ### Find the percentage rate

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...
32. ### Find the percent of the price

108.257=100(1+0.09)(1+p),p=0.0068, seems not right for me

yes
34. ### Find the percent of the price

For b, Lucy will rise the price by 0.08257?
35. ### Find the percent of the price

I think I get it, 1 = 1.09(1+p), p = 0.08257
36. ### Find the percent of the price

1 -1.09/1=0.09

0.09?
38. ### Find the percent of the price

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...
39. ### Prove Sup S = c

That is a typo
40. ### Prove Sup S = c

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...
41. ### Show that Q_F is not a division ring.

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...
42. ### Show √ 2 + √ 3 algebraic over Q

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...
43. ### FInd non-zero elements are primitive in a field

P (x)=x^4+1, we have p (0) mod 2 =1 and p (1)=2 which show p (x) is irreducible in Z_2, hence F_16 isomorphic with Z_2 [x]/p (x)
44. ### FInd non-zero elements are primitive in a field

x^2-2, If I didn't make mistakes, p (x)=x^2 - 3
45. ### FInd non-zero elements are primitive in a field

P (x) would be a maximal ideal and irreducible, I think x^3-x-1
46. ### FInd non-zero elements are primitive in a field

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...
47. ### Show F is isomorphic to F/{0}

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.
48. ### Show F is isomorphic to F/{0}

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...
49. ### Show the group of units in Z_10 is a cyclic group of order 4

Right, 1 can't generate the whole. So there is only 3 and 7 isomorphic with Z_4.
50. ### Show the group of units in Z_10 is a cyclic group of order 4

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...