The variables [math]S[/math] and [math]T[/math] have the joint density function
Determine [math]\operatorname{Cov}(T,S)[/math].
- -56.25
- 0
- 18.75
- 68.75
- 168.75
- Created by Admin, Jun 02'22
The variables [math]X[/math] and [math]Y[/math] have joint density function
with [math]\alpha \gt 1 [/math]. Determine the limit of the covariance of [math]X[/math] and [math]Y[/math] as [math]\alpha \rightarrow 1 [/math].
- -0.0525
- -1/12
- 0
- 0.0525
- 0.1775
- Created by Admin, Jun 02'22
Let [math]X,Y[/math] be any two random variables. Which of the following statements is always true:
- [math]|\operatorname{Cov}(X,Y)| \lt |\operatorname{Cov}(\operatorname{E}[X | Y],Y)|[/math]
- [math]|\operatorname{Cov}(X,Y)| \gt |\operatorname{Cov}(\operatorname{E}[X | Y],Y)|[/math]
- [math]\operatorname{Cov}(X,Y) = \operatorname{Cov}(\operatorname{E}[X | Y],Y)[/math]
- If [math]\operatorname{Cov}(\operatorname{E}[X | Y],Y) = 0[/math] then [math]X [/math] and [math]Y[/math] are independent.
- If [math]\operatorname{Cov}(X,Y) = \operatorname{Cov}(\operatorname{E}[X | Y],Y)[/math] for every [math]Y[/math] then [math]X = \operatorname{E}[X | Y][/math].
- Created by Admin, Jun 02'22
The daily stock returns [math]r_1[/math] and [math]r_2[/math] have identical marginal distributions with an expected return equal to zero. The returns are independent with a mean return of 0.05, given that both returns are less than -0.2 The covariance equals 0.0225 and the means equal -0.25, given that one return is greater than -0.2.
Determine the covariance of [math]r_1[/math] and [math]r_2[/math].
- 0
- 0.01317
- 0.01417
- 0.0755
- 0.0795
- Created by Admin, Jun 02'22
Let [math]X[/math] be a random variable that takes on the values –1, 0, and 1 with equal probabilities. Let [math]Y = X^2 [/math] . Which of the following is true?
- Cov(X, Y) > 0; the random variables X and Y are dependent
- Cov(X, Y) > 0; the random variables X and Y are independent
- Cov(X, Y) = 0; the random variables X and Y are dependent
- Cov(X, Y) = 0; the random variables X and Y are independent
- Cov(X, Y) < 0; the random variables X and Y are dependent
- Created by Admin, May 06'23
Let [math]X[/math] and [math]Y[/math] be continuous random variables with joint density function
Calculate the covariance of [math]X[/math] and [math]Y[/math].
- 0.04
- 0.25
- 0.67
- 0.80
- 1.24
- Created by Admin, May 06'23
[math]X[/math] and [math]Y[/math] denote the values of two stocks at the end of a five-year period. [math]X[/math] is uniformly distributed on the interval (0, 12). Given [math]X = x[/math], [math]Y[/math] is uniformly distributed on the interval [math](0, x)[/math].
Calculate [math]\operatorname{Cov}(X, Y)[/math] according to this model.
- 0
- 4
- 6
- 12
- 24
- Created by Admin, May 06'23
Let [math]X[/math] denote the size of a surgical claim and let [math]Y[/math] denote the size of the associated hospital claim. An actuary is using a model in which
- [math]\operatorname{E}[X] = 5 [/math]
- [math]\operatorname{E}[X^2] = 27.4 [/math]
- [math]\operatorname{E}[Y] = 7 [/math]
- [math]\operatorname{E}[Y^2] = 51.4 [/math]
- [math]\operatorname{Var}[X + Y] = 8 [/math]
Let [math]C_1 = X + Y [/math] denote the size of the combined claims before the application of a 20% surcharge on the hospital portion of the claim, and let [math]C_2[/math] denote the size of the combined claims after the application of that surcharge.
Calculate [math]\operatorname{Cov}(C_1,C_2)[/math] .
- 8.80
- 9.60
- 9.76
- 11.52
- 12.32
- Created by Admin, May 06'23
An actuary analyzes a company’s annual personal auto claims, [math]M[/math], and annual commercial auto claims, [math]N[/math]. The analysis reveals that [math]\operatorname{\operatorname{Var}}(M) = 1600 [/math], [math]\operatorname{Var}(N) = 900 [/math], and the correlation between [math]M[/math] and [math]N[/math] is 0.64.
Calculate [math]\operatorname{\operatorname{Var}}(M + N)[/math].
- 768
- 2500
- 3268
- 4036
- 4420
- Created by Admin, May 06'23
Points scored by a game participant can be modeled by [math]Z = 3X + 2Y – 5[/math]. [math]X[/math] and [math]Y[/math] are independent random variables with [math]\operatorname{Var} (X) = 3[/math] and [math]\operatorname{Var} (Y) = 4[/math].
Calculate [math]\operatorname{Var}(Z) [/math].
- 12
- 17
- 38
- 43
- 68
- Created by Admin, May 06'23