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multiple choice
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Suppose the random variables X and Y are independent of each other, and X~B(1,0.5), Y~U(-1,1), then P{X+Y≥0.5}=().
A. 0.5
B. 0.75
C. 0.25
D. 1
【Correct answer: A】 -
The known random variables X and Y are independent and identically distributed, and P(X=-1)=1-P(X=1)=p(0<p<1), let U = { 0 , XY = 1 ,
1 , XY = − 1. U= \begin{cases} 0, \quad XY=1, \\ 1, \quad XY=-1. \end{cases}U={ 0,XY=1,1,XY=−1.
The correct one of the following options is ().
A. P{U=0}=2p(1-p)
B. P{X=1,U=0}= p
C. When p = 1 2 p= \frac {1}{2}p=21, X and U are independent of each other.
D. P{X=1,U=1}= p
【Correct answer: C】 -
Assuming random variables X and Y are independent of each other, which of the following options is incorrect is ().
A. If X ~ B(n, p), Y ~ B(m, p), then X+Y ~ B(m+n, p) B. If X ~ P(λ), Y ~ P(μ
) , then X+Y ~ P(λ+μ)
C. If X ∼ N ( μ 1 , σ 1 2 ) , Y ∼ N ( μ 2 , σ 2 2 ) X \sim N( \mu _{1}, \sigma _{1}^{2}),Y \sim N( \mu _{2}, \sigma _{2}^{2})X∼N ( m1,p12),Y∼N ( m2,p22)，刪X + Y ∼ N ( μ 1 + μ 2 , σ 1 2 + σ 2 2 ) X+Y \sim N( \mu _{1}+ \mu _{2}, \sigma _{1} ^{2}+ \sigma _{2}^{2})X+Y∼N ( m1+m2,p12+p22)
D. If X ~ Exp(θ₁), Y~Exp(θ₂), then X+Y ~ Exp(θ₁+θ₂)
【Correct answer: D】 -
Suppose the random variable X and Y have the same distribution, and its distribution function is F(x), remember the distribution function of the random variable X+Y is G(x), then there is ().
A. G(2x)=2F(x)
B. 2G(2x)=F(x)
C. G(2x)≤2F(x)
D. G(2x)≥2F(x)
【Correct answer: C】 -
Let the density function of the two-dimensional random variable (X, Y) be f(x, y), then the two-dimensional random variable ( 3 X , Y 2 ) (3X, \frac {Y}{2})( 3 X ,2Y) density function
f ( 3 x , Y 2 ) ( u , v ) f_{(3x ,\frac {Y}{2})}(u,v)f( 3 x ,2Y)(u,v)=（）。
A. 3 2 f ( 3 u , v 2 ) \frac {3}{2}f(3u, \frac {v}{2}) 23f(3u,2v)
B. 2 3 f ( u 3 , 2 v ) \frac {2}{3}f \left ( \frac {u}{3} , 2v \right ) 32f(3u,2v)
C. f ( 3 u , v 2 ) f(3u, \frac {v}{2}) f(3u,2v)
D. f ( u 3 ; 2 v ) f \left ( \frac {u}{3};2v \right ) f(3u;2 v )
【Correct answer: B】