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 choiceLet 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)EX7.8) 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.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))²Suppose 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?Let 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 0Q1. 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