2. Covariance matrix of temperature measurements in a weather = station, represented with a random vector X=[X₁ X2 X3]', is given as: Cx It is known that RVs X₁ are random Gaussian in nature 14 12.1 10.87 12.1 12 0 L10.8 0 11 a. What is the variance in these three measurements? Hint: Remember Cov(X,X)=Var(X) b. What are the Cov[X₁,X2] and Cov[X3, X1]? c. Are X₁ independent from one another?
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- The random variables X,Y have variance Var(X)=36 and Var(Y)=1 and their correlation is Cor(X,Y)=−3/4. Calculate Var(X+Y) with a full explanationConsider the model Y = B1 + B2X + e, which we estimate using a random sample with 12 observations. Let bj and b2 be the estimators for B1 and B2 and recall Eê, where ê; = Y; – bị – b2X;. Suppose the sample correlation between {X; }"_-1 and {Y;}"_, is 0.5 and E(Y; – Ỹ„)² = 100. What is ổ?? %3D n-2 Hint: (i) for simple regression, the regression R? is equal to the squared sample correlation between X and Y. (ii) R² = 1 – SSE where SSE = Eế and SST SST = O a. 7.5 O b. 8.5 О с. 9 O d. 10 O e. 8 O f. 7 Clear my choiceEX7.8) Let Y be a random variable having a uniform normal distribution such that Y U(2,5) 2 Find the variance of random variable Y.
- Let I be an indicator variable with p(I) = {p , I=1 } {1-p , I=0}. Show that the variance of I is p(1-p)8. Given the Beta Distribution where p(x) = B(a,b).ªªa- Calculate the mean and variance B(a, b) = -xª-¹ (1-x)b-1 1 = √₁² 2²- x-¹(1-x)-¹dx = r(a) I (b) I(a + b)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.
- Show that variance o? = (x²) – ((x))²Let X and Y be identically distributed with mean 4 and variance 25.(a) If X and Y are uncorrelated, what is the variance of X + Y?(b) If the covariance between X and Y is 8, what is the variance of X + Y? (c) If X and Y are independent, what is E[(X - Y)^2]?1 The PDF of the random variable X is fx(x)= 6√x What are i) mean, ii) the mean square value and iii) the variance of random variable x. CS Scanned with CamScanner 0Suppose 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?Q1. By checking the correct or incorrect statements below into brackets, find the answer in (T) true, false (F) conditions. ) Variance population S is shown in example with o. () The calculations mean and variance over population elements are called parameter. ( ) The number of people injured in traffic accidents is continuously variable. () Patients with AIDS in a region is a discrete variable.Suppose that a random vector Y' = (Y1, Y2, Y3) has a multi-normal distribution with mean vector u and variance-covariance matrix E given by 1 2 and Σ= respectively. 1 1 What is the distribution of Z = Y? 1 1SEE MORE QUESTIONS