1. Suppose that {Yin: 1 ≤ i ≤n; n = 1,2,...} is a triangular array of random variables such that • for each n, Y₁,n.... Yn,n are independent and identically distributed; •√nE(Y1.n) → (with μ| <∞) and E (Y) → 0 as n→ ∞o. Use Chebyshev's (or Markov's) inequality to show that S₁ = n-¹/²₁Yi,np. i=1

Q: 4. Let X₁, i = 1,2,..., be independent random variables with P(X; = 1) = ½, P(X; = − 1) = 1. Put Fo=…

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Q: Exercise 13. Let X₁,..., Xn~ N(x, 02) i.i.d. and Y₁,..., Yn~ N(Hy, 02) i.i.d. with known o2. We…

A: Hi! Thank you for the question. As per the honor code, we are allowed to answer three sub-parts at a…

Q: Let a > 0 and X1, X2, ..., Xn, .. be a sequence of random variables such tha ..... 1 P(X, = 1) = 1 –…

A: Convergence in probability: Let X1,X2,.....Xn be the sequence of random variables converges in…

Q: 1. Suppose that the random variable Y₁, Y2, ..., Yn satisfy Yi = pi + Eis i=1,2,..., n, where x₁,…

A: denotes a random variable. denote fixed constants. are iid .

Q: c) Let X and Y be two discrete random variables. We define Z = X + Y , i.e. Vw E 2,Z(@) = X(@)…

A: Given that X and Y be two discrete random variables and the random variable Z=X+Y

Q: Suppose that X₁, X₂, X, are mutually independent random variables with the respective moment…

A: a) Given that Consider,

Q: Let {Xn} be iid random variables on R and suppose X₁ is not de- (constant) and EX² < ∞. Let Sn = 1…

A: From the given information, Xn1∞ be iid random variables on R. X1 is not deterministic and…

Q: Consider a memoryless discrete-time source which emits a sequence of random elements X1, X2,'· • ,…

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Q: random variables X, and Y, are correlated, with Corr(X₁, Y₁) € {1,...,n}. Finally, let X, Y and D be…

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Q: Exercise 13. Let X₁,..., Xn~ N(a,02) i.i.d. and Y₁,..., Yn~ N(y, o2) i.i.d. with known o². We assume…

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Q: 1. Show that the scaled symmetric random walk (W) (t): 0≤t≤T} is a martingale.

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Q: Exercise 13. Let X₁,..., Xn~ N(x, 02) i.i.d. and Y₁,..., Yn~ N(y, o2) i.i.d. with known o². We…

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Q: particular fruit's weights are normally distributed, with a mean of 466 grams and a standard…

A: P(443&lt;X&lt;528)=?

Q: Let Xi be independent Uniform(−6,6) random variables, for i=1, 2, ..., 13. Let Y = X1+X2+...+X13…

A: Given that X is uniformly distributed over the interval (-6,6). Therefore, the PDF of X is given…

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…

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Q: Q 6.3. Let X = (X1, X2,..., Xn) be a random vector with mean vector μ and variance- covariance…

A: Given that  be a random vector with mean vector  and variance-covariance matrix .

Q: 7. Let X₁, X2,..., Xn be independent variables, X; being N(µi, 1), and let y = = x² + x² +...+X²/2.…

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Q: 1. X1, X2, and X3 are i.i.d. random variables with each X; ~ N(0, o²). Let Z and Y be defin ed as Z…

A: It is an important part of statistics. It is widely used.

Q: Let X and Y be independent random variables, both being equally likely to be any of the values 1, 2,…

A: The  and  are independent random variables. and takes equally likely to be any of the values .

Q: Let Xi be independent Uniform(−6,6) random variables, for i=1, 2, ..., 13.  Let Y = X1+X2+...+X13 .…

A: The objective of the question is to determine the correct statements about the moment generating…

Q: Suppose X - N(u, o?) where o is known. Let X1, X2,.... X, be a random sample of size n. Prove that u…

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Q: 3. Suppose that discrete random variables X and Y have the following incomplete joint distri- bution…

A: Farmula used  V(X+Y)=V(X)+V(Y)-2Cov(X,Y)

Q: Suppose that X1,X2,... are iid random variables, each of which is equal to zero with probability ,…

A: The distribution of X is given as X 0 1 P(X=x) 23 13 Calculating the expected value of X…

Q: 16 3. Let X₁, X2,..., Xn be random variables with a common mean μ. Sup- pose that cov[X₁, X;] = 0…

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Q: We say that a mean-zero random variable X is sub-Gaussian with parameter o2 if the m.g.f. of X…

A: Markov Inequality Markov's inequality gives an upper boundary for the probability that a…

Q: Suppose X is a random variable taking possible values in {1, 2, 3,... } and 0 < P(X = 1) < 1, and…

A: Let X be the random variable having possible values {1,2,3...} and 0&lt;p&lt;1X satisfies the…

Q: Let X₁, i = 1,2,...,n be mutually uncorrelated random vectors of same size and with E(X;) = µ¡. Show…

A: The problem asks us to show that the expected value of the norm squared of the sum of the…