1. Simple Random Walk Let (Sn)n≥o be a simple random walk starting at 1(So = 1) and with p = 0.3 and q = 1−p = 0.7. Compute the following probabilities: • P(S₁ = 0|S5 = 0), • P(S5 0|S32), = ● P(M10 ≥ 4, S10 ≥ 4), where M10 = max0≤i≤10 Si.

Q: 2. Let X be a random variable. Is it possible that px,x² = 1 for some X? What about PX,x² = -1?

A: Solution

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A: We have given in the question: P(30≤X≤79)=0.26P(X>79)=0.17

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A:

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A: Solution: 3. It is needed to prove that

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Q: X y 46.6 156.6 43.7 71.6 49.5 57.4 -0.6 -12.3 17.2 25.7 55.2 39.1 -0.7 27.5 23.1 35.4 34.5 42.3 24.8…

A: Given:

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Q: For any two random variables Y₁ and Y₂, Var(Y₁Y₂|Y₁ = y₁) = y{Var(Y₂|Y₁ = y₁). TRUE

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A:

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Q: please solve it on paper

A:

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A:

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Q: 2. 0) with p = 0.4 Compute the = Let (Sn)nzo be a simple random walk starting in 0 (i.e. So : 1 − p…

A:

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Q: (a) The correlation

A: Answer:

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A: Farmula used  Range=Maximum-Minimum

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Q: What is the pmf of X? Calculate the expectation of X.

A: The moment generating function of random variable X is defined as mX(t) = E(etX)