8.4 （a） Q 8.4. Let X₁, X2, Y₁ and Y2 be independent random variables each having a Gaussian dis-tribution. Suppose that E[X₁] = E[X₂] = µ₁, that E[Y₁] = E[Y₂] = μ2 and that var(X₁) =var (X₂) = var(Y₁) = var (Y₂) = o² > 0.Let X = (X₁ + X₂) and Ỹ = ½ (Y₁ + Y₂). Define a random variable S² via15-2The aim of this question is to describe the joint distribution of X - Y and S²Σ(Xi - X)² + (Y; - Y)²+2₁²-1²)i=1i=1(a) Let U be a 4 × 4 matrix whose first two rows are(1/√21/√2 1/√√2 00 1/√21/√₂)1/√2)Choose two further rows so that U is an orthogonal matrix. [Hint: it's a good plan to useplenty of zeros!]

Given the first two rows of matrix U:

Answered: (a) Let U be a 4 x 4 matrix whose first…

Trigonometry (MindTap Course List)

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ISBN:9781337278461

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Chapter6: Topics In Analytic Geometry

Section6.4: Hyperbolas

Problem 5ECP: Repeat Example 5 when microphone A receives the sound 4 seconds before microphone B.

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Repeat Example 5 when microphone A receives the sound 4 seconds before microphone B.
B) Let X1,X2, .,Xn be a random sample from a N(u, o2) population with both parameters unknown. Consider the two estimators S2 and ô? for o? where S2 is the sample variance, i.e. s2 =E,(X, – X)² and ở² = 'E".,(X1 – X)². [X = =E-, X, is the sample mean]. %3D n-1 Li%3D1 [Hint: a2 (п-1)52 -~x~-1 which has mean (n-1) and variance 2(n-1)] i) Show that S2 is unbiased for o2. Find variance of S2. ii) Find the bias of 62 and the variance of ô2. iii) Show that Mean Square Error (MSE) of ô2 is smaller than MSE of S?. iv) Show that both S2 and ô? are consistent estimators for o?.
Suppose X and Y are independent. X has a mean of 1 and variance of 1, Y has a mean of 0, and variance of 2. Let S=X+Y, calculate E(S) and Var(S). Let Z=2Y^2+1/2 X+1 calculate E(Z). Hint: for any random variable X, we have Var(X)=E(X-E(X))^2=E(X^2 )-(E(X))^2, you may want to find EY^2 with this. Calculate cov(S,X). Hint: similarly, we have cov(Z,X)=E(ZX)-E(Z)E(X), Calculate cov(Z,X). Are Z and X independent? Are Z and Y independent? Why? What about mean independence?
Q 8.4. Let X₁, X2, Y₁ and Y₂ be independent random variables each having a Gaussian dis- tribution. Suppose that E[X₁] = E[X₂] = µ₁, that E[Y₁] = E[Y₂] = µ2 and that var(X₁) var (X₂) = var(Y₁) = var(Y₂) = o² > 0. Let X = (X₁ + X₂) and Ỹ = ½ (Y₁ + Y₂). Define a random variable S² via 8² - 1 (21X₁-RP+ (M-91²) 82 = 2 2 i=1 i=1 The aim of this question is to describe the joint distribution of X – Ỹ and S² (a) Let U be a 4 x 4 matrix whose first two rows are (1/1/2 √√2 1/√2 0 0 1/√ 2 1/√2) = Choose two further rows so that U is an orthogonal matrix. [ Hint: it's a good plan to use plenty of zeros!] (b) Let W be the 4-dimensional random vector W = (X₁, X2, Y₁, Y₂), and define V to be the random vector V = UW. Find the mean vector and variance-covariance matrix of V. (c) Show that you can write S² as a function of V3 and V4 and that Ă – Ỹ is a function of V₁ and V₂. Use this to describe the joint distribution of X – Y and S².
5. Let Y,, Y2, ., Yn be independent, exponentially distributed random variables with mean 0/2. Show that the variance of the minimum, Y1) = min(Y,, Y2, ...,n), are E(Y1)) Var(Ya)) = and 2n 02 4n²°
If X₁, X2,..., Xn constitute a random sample of size n from an exponential population, show that X is a sufficient estimator of the parameter 0.
Consider two independent exponential random variables X1 and X2 with parameter lambda=1. LetY1 = X1 Y2 = X1 + X2. Find the MMSE estimate of Y1 using Y2.
2. Y1, Y2, ..., Yn are i.i.d. exponential random variables with E{Yi} = 1/θ. Find thedistribution of Y =1 nPiYi.
4. Suppose that X has pdf f(x) = 3x² for 0 < x< 1. Find the pdf of the random variable Y = VX.
Q 8.4. Let X₁, X2, Y₁ and Y2 be independent random variables each having a Gaussian dis- tribution. Suppose that E[X₁] = E[X₂] µ₁, that E[Y₁] = E[Y₂] μ2 and that var(X₁) var (X₂) = var (Y₁) = var(Y₂) = o² > 0. Let X = (X₁ + X₂) and Ỹ = ½ (Y₁ + Y₂). Define a random variable S² via S² = 2 Σ(Xi − X)² + Σ(Y₂ − Ỹ)² ΣΥ i-n²) i=1 1 (2₁x i=1 = (1/√2 1/√2 0 The aim of this question is to describe the joint distribution of X - Y and S² (a) Let U be a 4 4 matrix whose first two rows are 0 0 1/√2 1/√₂) - Choose two further rows so that U is an orthogonal matrix. [Hint: it's a good plan to use plenty of zeros!] (b) Let W be the 4-dimensional random vector W = (X₁, X2, Y₁, Y2), and define V to be the random vector V = UW. Find the mean vector and variance-covariance matrix of V. (c) Show that you can write S² as a function of V3 and V4 and that X - Y is a function of V₁ and V₂. Use this to describe the joint distribution of X - Y and S².
2.5.8) Suppose that Y is a continuous random variable whose pdf is given by f(y) = {K (4y = 2y²), 0 1). (c) Find F(y).
Let Y₁, Y2, ..., Yn denote a random sample of size n from a population with a uniform distribution on the interval (0,0). Consider = Y(1) = min(Y₁, Y₂, ..., Yn) as an estimator for 0. Show that is a biased estimator for 0.

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(a) Let U be a 4 x 4 matrix whose first two rows are (1/√/2 0 (1/√2 1/√2 0 0 1/√2 1/√2, 1/√₂) Choose two further rows so that U is an orthogonal matrix. [Hint: it's a good plan to use plenty of zeros!]