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Mathematical Statistics (I) FINAL EXAM Spring, 2011 1. Let \( X_{1}, \cdots, X_{k+1} \) be independent random variables, each having a gamma distribution with \( \beta=1 \), that is, the pdf of \( X_{i} \) is \( f_{X_{i}}\left(x_{i}\right)=\frac{1}{\Gamma\left(\alpha_{i}\right)} x_{i}^{\alpha_{i}-1} e^{-x_{i}} I\left(0<x_{i}<\infty\right) \). Let \[ Y_{i}^{\prime}=\frac{X_{i}}{X_{1}+\cdots+X_{k+1}}, i=1, \cdots, k . \] o (a) Find the joint pdf of \( Y_{1}, \cdots, Y_{k} \). (b) What is the distribution of \( Y_{1}+\cdots+Y_{r} \), where \( r \leq k \) ? \( n \cdot( \) \( x .2 .24 \) \( x \) \( x \) 2. Let \( f(x) \) and \( F(x) \) be the pdf and the cdf of a distribution of the continuous type such that \( f^{\prime}(x) \) exists for all \( x \). We say that the random variable \( Y \) has a truncated distribution, if \( Y \) has pdf \( g(y)=f(y) / F(b) I(-\infty<y<b) \). Suppose that \( E(Y)=-f(b) / F(b) \) for all real \( b \). Prove that \( f(x) \) is the pdf of standard normal distribution. 3. Let \( X \) have an exponential distribution. O (a) Show that \( P(X>x+y \mid X>x)=P(X>y) \) for any positive \( x \) and \( y \). (This property is called the memoryless property). o(b) Let \( F^{\prime} \) be the cdf of a continuous random variable \( Y \). Suppose that \( F \) has the memoryless property and that \( F(0)=0 \) and \( 0<F(y)<1 \) for \( y>0 \). Show that \( F(y)=1-e^{-\lambda y} \) for \( y>0 \) for some \( \lambda>0 \). 4. Let \( X_{n} \) converges in distribution to \( X \) and \( Y_{n} \) converges in probability to \( c \) (a constant). Show that o (a) \( X_{n}+Y_{n} \) converges in distribution to \( X+c \). (b) \( X_{n} Y_{n} \) converges in distribution to \( c X \). 5. Let \( X_{n} \sim \operatorname{Poi}(n \mu) \). (a) Find the limiting distribution of \( \frac{\sqrt{n}\left(\frac{x_{n}}{n}-\mu\right)}{\sqrt{\mu}} \). (b) Find a 'variance stabilizing transform', that is, find a transform \( h \) such that \( h\left(X_{n} / n\right) \) has asymptotic variance which is independent of \( \mu \). 6. Let \( X_{1}, \cdots, X_{n} \) be a random sample from \( N\left(\mu, \sigma^{2}\right) \) and let \( S^{2}=\frac{1}{n-1} \sum_{i=1}^{n}\left(X_{i}-\bar{X}\right)^{2} \) 6. Let \( X_{1}, \cdots, X_{n} \) be a random sample from \( N\left(\mu, \sigma^{2}\right) \) and let \( S^{2}=\frac{1}{n-1} \sum_{i=1}^{n}\left(X_{i}-\bar{X}\right)^{2} \) 1

          Mathematical Statistics (I)
FINAL EXAM
Spring, 2011
1. Let \( X_{1}, \cdots, X_{k+1} \) be independent random variables, each having a gamma distribution with \( \beta=1 \), that is, the pdf of \( X_{i} \) is \( f_{X_{i}}\left(x_{i}\right)=\frac{1}{\Gamma\left(\alpha_{i}\right)} x_{i}^{\alpha_{i}-1} e^{-x_{i}} I\left(0<x_{i}<\infty\right) \). Let
\[
Y_{i}^{\prime}=\frac{X_{i}}{X_{1}+\cdots+X_{k+1}}, i=1, \cdots, k .
\]
o (a) Find the joint pdf of \( Y_{1}, \cdots, Y_{k} \).
(b) What is the distribution of \( Y_{1}+\cdots+Y_{r} \), where \( r \leq k \) ?
\( n \cdot( \)
\( x .2 .24 \)
\( x \)
\( x \)
2. Let \( f(x) \) and \( F(x) \) be the pdf and the cdf of a distribution of the continuous type such that \( f^{\prime}(x) \) exists for all \( x \). We say that the random variable \( Y \) has a truncated distribution, if \( Y \) has pdf \( g(y)=f(y) / F(b) I(-\infty<y<b) \). Suppose that \( E(Y)=-f(b) / F(b) \) for all real \( b \). Prove that \( f(x) \) is the pdf of standard normal distribution.
3. Let \( X \) have an exponential distribution.
O (a) Show that \( P(X>x+y \mid X>x)=P(X>y) \) for any positive \( x \) and \( y \). (This property is called the memoryless property).
o(b) Let \( F^{\prime} \) be the cdf of a continuous random variable \( Y \). Suppose that \( F \) has the memoryless property and that \( F(0)=0 \) and \( 0<F(y)<1 \) for \( y>0 \). Show that \( F(y)=1-e^{-\lambda y} \) for \( y>0 \) for some \( \lambda>0 \).
4. Let \( X_{n} \) converges in distribution to \( X \) and \( Y_{n} \) converges in probability to \( c \) (a constant). Show that
o (a) \( X_{n}+Y_{n} \) converges in distribution to \( X+c \).
(b) \( X_{n} Y_{n} \) converges in distribution to \( c X \).
5. Let \( X_{n} \sim \operatorname{Poi}(n \mu) \).
(a) Find the limiting distribution of \( \frac{\sqrt{n}\left(\frac{x_{n}}{n}-\mu\right)}{\sqrt{\mu}} \).
(b) Find a 'variance stabilizing transform', that is, find a transform \( h \) such that \( h\left(X_{n} / n\right) \) has asymptotic variance which is independent of \( \mu \).
6. Let \( X_{1}, \cdots, X_{n} \) be a random sample from \( N\left(\mu, \sigma^{2}\right) \) and let \( S^{2}=\frac{1}{n-1} \sum_{i=1}^{n}\left(X_{i}-\bar{X}\right)^{2} \)
6. Let \( X_{1}, \cdots, X_{n} \) be a random sample from \( N\left(\mu, \sigma^{2}\right) \) and let \( S^{2}=\frac{1}{n-1} \sum_{i=1}^{n}\left(X_{i}-\bar{X}\right)^{2} \)
1

Mathematical Statistics (I)
FINAL EXAM
Spring, 2011
1. Let X1, ⋯, Xk+1 be independent random variables, each having a gamma distribution with β=1, that is, the pdf of Xi is fXi(xi)=(1)/(Γ(αi)) xi^αi-1 e^-xi I(0<xi<∞). Let

Yi^'=(Xi)/(X1+⋯+Xk+1), i=1, ⋯, k .

o (a) Find the joint pdf of Y1, ⋯, Yk.
(b) What is the distribution of Y1+⋯+Yr, where r ≤ k ?
n ·(
x .2 .24
x
x
2. Let f(x) and F(x) be the pdf and the cdf of a distribution of the continuous type such that f^'(x) exists for all x. We say that the random variable Y has a truncated distribution, if Y has pdf g(y)=f(y) / F(b) I(-∞<y<b). Suppose that E(Y)=-f(b) / F(b) for all real b. Prove that f(x) is the pdf of standard normal distribution.
3. Let X have an exponential distribution.
O (a) Show that P(X>x+y | X>x)=P(X>y) for any positive x and y. (This property is called the memoryless property).
o(b) Let F^' be the cdf of a continuous random variable Y. Suppose that F has the memoryless property and that F(0)=0 and 0<F(y)<1 for y>0. Show that F(y)=1-e^-λ y for y>0 for some λ>0.
4. Let Xn converges in distribution to X and Yn converges in probability to c (a constant). Show that
o (a) Xn+Yn converges in distribution to X+c.
(b) Xn Yn converges in distribution to c X.
5. Let Xn∼Poi(n μ).
(a) Find the limiting distribution of (√(n)((xn)/(n)-μ))/(√(μ)).
(b) Find a 'variance stabilizing transform', that is, find a transform h such that h(Xn / n) has asymptotic variance which is independent of μ.
6. Let X1, ⋯, Xn be a random sample from N(μ, σ^2) and let S^2=(1)/(n-1)∑i=1^n(Xi-X̅)^2
6. Let X1, ⋯, Xn be a random sample from N(μ, σ^2) and let S^2=(1)/(n-1)∑i=1^n(Xi-X̅)^2
1

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