4. Suppose X₁ and X₂ are independent random variables with cdf Fx (x) = sin(x), 0 ≤ x ≤ 1/ a. Show that fx(2) (x) = 2sin(x) cos(x), where X (2) is the random variable for the 2nd order statistic. (Remember, we only have 2 random variables.) . b. Show that fx) (r) = 2cos(r)[1 - sin(x)], where X(1) is the random variable for the 1st order statistic.
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![4. Suppose X₁ and X₂ are independent random variables with cdf Fx(x) = sin(x), 0 ≤ x ≤ 1/
a. Show that fx(2) (x) = 2sin(x) cos(x), where X (2) is the random variable for the 2nd order statistic.
(Remember, we only have 2 random variables.)
b. Show that fx(1)(x) = 2cos(r) [1 sin(x)], where X(1) is the random variable for the 1st order
statistic.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffc2a3207-1546-4f7c-866f-8a9e0d862ab8%2Fcca7d9dc-27d1-4da2-8fce-45a13f825531%2Fjgikiif_processed.jpeg&w=3840&q=75)
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- Repeat Example 5 when microphone A receives the sound 4 seconds before microphone B.Suppose X and Y are two random variables with E[X] = 1, Var (X) = 4, E[Y ] = -1, Var (Y) 4, and Cov (X, Y) = 1. Find the standard deviation of (X - Y). = (a) 2 (b) √2 (c) 6 (d) √6 (e) None of the aboveB) 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?4. Suppose that X has pdf f(x) = 3x² for 0 < x< 1. Find the pdf of the random variable Y = VX.1. Random variable X has p.d.f be 6z I> 0 fx(z) = || otherwise and let Y = eX. (b) (i.e. Calculate Var(Y)) Calculate the variance of Y.
- 3. Consider two random variables X₁ and X2 whose joint pdf is given by for x₁0, x2 > 0, x₁ + x2 < 2, { Find the pdf of Y = X₂ - X₁. f(x1, x₂) = NIT otherwise.16) A stationary random process has an autocorrelation function of Rx (T)=16-e cos 20rt+8 cos10TT. Find the variance of this process.4. Assume that X is an exponential random variable. Suppose further that Var(X) = 5. E(X). (a) Find the parameter 0> 0. (b) Compute Var(X + 4). (c) Compute P(X > 15[X > 11).
- B) Let the random variable X have the moment generating function e3t M(t) for -1Prove that the variance of b, V(b) = o²(X'X)−¹21. Let the random variable X have the moment generating function e3t M(t) : -1SEE MORE QUESTIONSRecommended textbooks for youTrigonometry (MindTap Course List)TrigonometryISBN:9781337278461Author:Ron LarsonPublisher:Cengage Learning
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