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  1. HaLAA

    Roll a fair die 5 times

    Yes, j,k,a,b gives the same outputs, we have 4 rolls have the same outputs.
  2. HaLAA

    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. HaLAA

    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. HaLAA

    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. HaLAA

    Roll a fair die 5 times

    I mean _ _ _... I _ _ _..._ j _ _ _...._ _k... something like this.
  6. HaLAA

    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. HaLAA

    Roll a fair die 5 times

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

    Roll a fair die 5 times

    I can have a combination of any 3 number of 6 if I understand correctly
  9. HaLAA

    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
  10. HaLAA

    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...
  11. HaLAA

    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
  12. HaLAA

    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 $x$...
  13. HaLAA

    Conditional Probability and law of total probability

    I never learn how to draw a tree
  14. HaLAA

    Conditional Probability and law of total probability

    I don't see how to draw probability tree.
  15. HaLAA

    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...
  16. HaLAA

    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...
  17. HaLAA

    Find all irreducible polynomials over F of degree at most 2

    α^2=α+1,(α+1)^2=α, the ch(F)=2
  18. HaLAA

    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...
  19. HaLAA

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

    At x =0, we still have f'(0)=0?
  20. HaLAA

    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
  21. HaLAA

    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...
  22. HaLAA

    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?
  23. HaLAA

    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]
  24. HaLAA

    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...
  25. HaLAA

    Find the percentage rate

    Thank you for checking my answer and help me to understand the question in more detail
  26. HaLAA

    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.
  27. HaLAA

    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
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