[Probability Theory] Variance and Standard Deviation of Random Variables Homework

multiple choice

1. Suppose the random variable X obeys the uniform distribution on the interval (-1, 2), then D(X)=()
   A.1/4
   B.1/2
   C.3/4
   D.1/8
   【Correct answer: C 】

2. Put red, yellow and white balls into two cups at random, if X is the number of cups with balls, then D(X)=()
   A 3/16
   B 13/4
   C 3/4
   D1
   【Correct answer: A】

3. Let the probability density of random variable X be
   $f(x)=\{\begin{array}{ll}1+x,& -1\le x<0,\\ 1-x,& 0<x\le 1,\\ 0,& Other\end{array}$
   Then variance DX=()
   A.1/3
   B.1/4
   C.1/5
   D.1/6
   【Correct answer: D】

4. Let random variable X~U(-1,2), let
   $Y=\{\begin{array}{ll}1,& X>0,\\ 0,& X=0,\\ -1,& X<0.\end{array}$
   Then D(Y)=（）
   A 2/3
   B 5/6
   C 5/9
   D 8/9
   【Correct answer: D】

5. Suppose the expectation E(X)=μ of the random variable x, and the variance D(X)=σ²>0, then there must be () for any constant c
   A. B(Xc)²=E(X²)-c².
   B. E (Xc)²=E(X-μ)²
   C. E(Xc)²<E(X-μ)².
   D. E(Xc)²≥E(X-μ)².
   【Correct answer: D】

6. Let the density function of random variable X be
   $f(x)=\{\begin{array}{ll}4x{e}^{-2x},& x>0,\\ 0,& others.\end{array}$
   Then D(-2X-2) = ()
   A. 0
   B. 1
   C. 2
   D. 4
   【Correct answer: C】

7. Let the random variables X and Y be independent of each other, and $X\sim U(0,2)$, $Y\sim U(0,1)$ , then ()
   A.$D(X-Y)=\frac{1}{4}$
   B. $D(X-2Y)\_=0$
   C. $D(X-Y)=\frac{1}{3}$
   D. $D(X-2Y+1)=\frac{2}{3}$
   【Correct answer: D】