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multiple choice
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It is known that F₁(x) and F₂(x) are distribution functions, if aF₁(x)-bF₂(x) is also a distribution function, then the correct one of the following options about constants a and b is ().
Aa= 3 5 \frac{3}{5}53,b= − 2 5 -\frac{2}{5} −52
B.a= 2 3 \frac{2}{3} 32,b= 2 3 \frac{2}{3} 32
C.a= − 1 2 -\frac{1}{2} −21,b= 3 2 \frac{3}{2} 23
D.a= 1 2 \frac{1}{2} 21;b= − 3 2 −\frac{3}{2} −23
【Correct answer: A】 -
Let the distribution function of random variable X be
f ( x ) = { 0 , x < 0 0.5 , 0 ≤ x < 1 1 − e − x , x ≥ 1 f(x) =\begin{cases} 0, & x < 0\\ 0.5, & 0≤x<1\\ 1-e^{-x}, & x ≥1 \end{cases}f(x)=⎩⎪⎨⎪⎧0,0.5,1−e−x,x<00≤x<1x≥1
Then P{X =1}=().
A.0.5
B.1−e⁻¹
C.0.5−e⁻¹
D. e⁻¹
【Correct answer: C】 -
The distribution function of the known non-degenerate random variable X is
F ( x ) = { 0 , x < 0 , a + be − x 2 2 , x ≥ 0. F(x)= \begin{cases} 0,&x<0 , \\ a+be^{- \frac {x^{2}}{2}},&x \ge 0. \end{cases}F(x)={ 0,a+be−2x2,x<0,x≥0.
Then there are ().
A. a=0, b=1
B. a=1, b=0
C. a=1, b=1
D. a=1, b=-1
【Correct answer: D】 -
Let the distribution function of random variable X be
F ( x ) = { 0 , x ≤ a , x − b , a < x ≤ 2 c , x > 2. F(x)= \begin{cases} 0,&x \le a, \\ xb,&a<x \le 2 \\ c,&x>2. \end{cases}F(x)=⎩⎪⎨⎪⎧0,x−b,c,x≤a,a<x≤2x>2.
Then P{X>1.5}=().
A. 0.5
B. 1
C. 0
D. 0.8
【Correct answer: A】 -
Which of the following functions is a distribution function is ().
A. F ( x ) = { 0 , x < 0 , e − z , x ≥ 0. F(x)= \begin{cases} 0,&x<0, \\ e^{-z},&x \ge 0. \end{cases}F(x)={ 0,e−z,x<0,x≥0.
B. F ( x ) = { 0 , x < 0 , 0.5 , 0 ≤ x ≤ 1 , 1 , x > 1. F(x)= \begin{cases} 0,&x<0, \\ 0.5,&0 \le x \le 1, \\ 1,&x>1. \end{cases} F(x)=⎩⎪⎨⎪⎧0,0.5,1,x<0,0≤x≤1,x>1.
C. F ( x ) = { 0 , x < 0 , 1 − x 2 , 0 ≤ x < 1 , 1 , x ≥ 1. F(x)= \begin{cases} 0,&x<0, \\ \frac {1-x}{2},&0 \le x<1, \\ 1,&x \ge 1. \end{cases} F(x)=⎩⎪⎨⎪⎧0,21−x,1,x<0,0≤x<1,x≥1.
D. F ( x ) = { 0 , x < 0 , sin x , 0 ≤ x ≤ π / 2 , 1 , x > π / 2. F(x)= \begin{cases} 0,&x<0, \\ \sin x,&0 \le x \le \pi /2, \\ 1,&x> \pi /2 \end{cases}F(x)=⎩⎪⎨⎪⎧0,sinx,1,x<0,0≤x≤p / 2 ,x>p / 2 .
【Correct answer: D】