数学概率作业代写 数学概率代写 数学作业代写 概率作业代写

4. Probability on finite sample spaces.

数学概率作业代写 1. Are the following events A, B ⊂ Ωroulette independent? Find P(A|B) and P(B|A) in each case. a) A = Red, B = Even,

1. Are the following events A, B ⊂ Ω_roulette independent? Find P(A|B) and P(B|A) in each case.

a) A = Red, B = Even,

b) A = {1, 2, 3, 4}, B = {1, 2, 3, 4, 5, 6, 7, 8, 9},

c) A = {1, 2, 3, 4, 5}, B = {6,7,8}.

2. A generalisation of 1b). Suppose A ⊂ B. Can A, B be independent?

3. A generalisation of 1c). Suppose A ∩ B = ∅. Can A, B be independent?

7. A die is thown twice. What is the probability that the difference: the first number minus the second number, is 0 or 1. What is the conditional probability of such an event if the die shows 6 at the first time.

8. We play a game where we have two identical boxes, one contains a coin and the other a die. We select a box and throw the device we find in the box. We win if the coin (if selected) shows Heads, or if the die shows 3 or more. What is the probability of winning?

9. (optional) 数学概率作业代写

[The following exercise is known as the Monty Hall problem. We give the original formulation, but before googling it, have a go.]

Suppose you’re on a game show, and you’re given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what’s behind the doors, opens another door, say No. 3, which has a goat. He then says to you, “Do you want to change you decision and pick door No. 2?” Which of the following statements is correct:

a. It is better to stick to the original choice. (What is the probability of winning the car then?)

b. You should switch to door No. 2. (What is the probability of winning the car then?)

c. It does not matter, the probabilities are equal.

10. (optional) 数学概率作业代写

Prove that if Ω has N elements with uniform probability, where N is a prime number, it is impossible to fifind independent non-trivial events (by trivial we mean the empty set or the whole Ω).

11. Find the distribution P_X and the cumulative distribution function F_X of X defined on Ω_die by X(ω) = ω(6 − ω). Find the expectation of X.

12. A random variable X is defined on Ωroulette by X(ω) = min{6, max{2ω−10, 0}}. Find the cumulative distribution function.

15. Consider two events A, B ⊂ Ω, and define X(ω) = 1_A(ω), Y (ω) = 1_B(ω). Prove that X, Y are independent if and only if A, B are independent.

16. 数学概率作业代写

On Ω_roulette consider two random variables: X(ω) = 1 for all ω ∈ Ω, and Y (ω) = (1 − ω)² . Compute the covariance Cov(X, Y ). Are X and Y independent? Is the form of Y relevant, that is, will the answer be different for some other Y ?

17. Let Ω = Ω_die × Ω_die = {(ω₁, ω₂) : ω₁ = 1, 2, 3, 4, 5, 6; ω₂ = 1, 2, 3, 4, 5, 6} and consider two random variables X(ω₁, ω₂) = ω₁ + ω₂, Y (ω₁, ω₂) = ω₁ − ω₂. Compute Cov(X, Y ). Are they independent?

18. On Ω = Ω_die × Ω_die = {(ω₁, ω₂) : ω₁ = 1, 2, 3, 4, 5, 6; ω₂ = 1, 2, 3, 4, 5, 6} consider two random variables X(ω₁, ω₂) = 2ω₁ − 3, Y (ω₁, ω₂) = 3 − ω₂. Are they independent?

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