[Probability Theory] Marginal Distribution Homework

multiple choice

1. Suppose the random variables X₁, X₂, X₃, X₄ have the same distribution and all obey B(1,p), then there is ().
   A. X₁+X₂ is identically distributed with X₃+X₄.
   B. X₁-X₂ is identically distributed with X₃-X₄.
   C. (X₁,X₂) is identically distributed with (X₃,X₄).
   D. $X_1^2,X_2^2,X_3^2,X_4^2$Same distribution.
   【Correct answer: D】

2. Let the density function of a two-dimensional random variable (X, Y) be
   $f(x,y)=\{\begin{array}{ll}c,& x^2\le y\le x,\\ 0,& others.\end{array}$
   Where c is a non-negative constant, the correct one of the following options is ().
   A. The constant c=4
   B. The edge density of x is $f_x(x)=\{\begin{array}{ll}x-x^2,& 0\le x\le 1,\\ 0,& others.\end{array}$
   C. P { X > 0.5 } = 1 2 P \{ X>0.5 \} = \frac {1}{2} P{ X>0.5}=21​
   D. $The\; edge\; density\; of\; Yis{f}_Y(y)=\{\begin{array}{ll}6(\sqrt{y}-y),& x^2\le y\le x,\\ 0,& others.\end{array}$
   【Correct answer: C】

3. Let the density function of a two-dimensional random variable (X, Y) be
   $f(x,y)=\{\begin{array}{ll}cxy,& 0\le x\le 1,1\le y\le 3,\\ 0,& others.\end{array}$
   where c is a non-negative constant. The correct one of the following options is ().
   A. Constant c=2
   B. P { X > 0.5 } = 1 2 P \{ X>0.5 \} = \frac {1}{2}P{ X>0.5}=21​
   C. $P\{AND>2\}=\frac{1}{2}$
   D. The function of Y functionf Y ( y ) = { y 4 , 1 ≤ y ≤ 3 0 , others . f_{Y}(y)= \{\begin{array}{l}\frac{y}{4},&1\le and\le3\\0,&\others.\end{array}fY​(y)={ 4y​,0,​1≤y≤3 others.​
   【Correct answer: D】

4. Let the distribution function of a two-dimensional random variable (X, Y) be F(x, y), and the marginal distributions are $F_x(x),F_y\left(y\right)$ , then the probability P{X>x, Y>y}=().
   A.$1-F(x,y)$
   B.$1-F_X(x)-F_Y(y)$
   C.$F(x,y)-F_X\left(x\right)-F_Y(y)+1$
   D.$F(x,y)+F_X\left(x\right)+F_Y\left(y\right)-1$
   [Correct answer: C]