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EXERCISE 1. Suppose Xi's are iid Negative Binomial(3,1/4) (a) Compute P(X1 < 5); (b) approximate P(21.9...
Suppose Xi's are independent and identically distributed Negative Binomial(3, 1/4): compute P(X1 <= 5).
Suppose Xi's are independent and identically distributed Negative Binomial(3, 1/4): compute P(X1 <= 5).
1. Suppose that X, X, X, are iid Berwulli(p),0 <p<1. Let U. - x Show that,...
1. Suppose that X, X, X, are iid Berwulli(p),0 <p<1. Let U. - x Show that, U, can be approximated by the N (np, np(1-P) distribution, for large n and fixed <p<1. 2. Suppose that X1, X3, X. are iid N ( 0°). Where and a both assumed to be unknown. Let @ -( a). Find jointly sufficient statistics for .
Let X1,X2 be two independent exponential random variables with λ=1, compute the P(X1+X2<t) using the joint...
Let X1,X2 be two independent
exponential random variables with λ=1, compute the
P(X1+X2<t) using the joint density function. And let Z be gamma
random variable with parameters (2,1). Compute the probability that
P(Z < t). And what you can find by comparing P(X1+X2<t) and
P(Z < t)? And compare P(X1+X2+X3<t) Xi iid
(independent and identically distributed) ~Exp(1) and P(Z < t)
Z~Gamma(3,1) (You don’t have to compute)
(Hint: You can use the fact that Γ(2)=1,
Γ(3)=2)
Problem 2[10 points] Let...
(4) Suppose that {X;}-1 iid random variables from a Binomial distribution Bin(m, p). Using your answer...
(4) Suppose that {X;}-1 iid random variables from a Binomial distribution Bin(m, p). Using your answer in (3) obtain an approximate 99% confidence interval for the pa- rameter p based on the MLE. Explain how you would estimate the Fisher information matrix.
5.2.5 (Example 5.2.6 Continued) Suppose thatXY are iid having the following common distribution. PC,-1-cip.i-1. 2. 3....
5.2.5 (Example 5.2.6 Continued) Suppose thatXY are iid having the following common distribution. PC,-1-cip.i-1. 2. 3. and 2 <p < 3 Here. c c(p) (> 0) is such that Σ | P(X,-i) = 1.. Is there a real number a = a(p) such that Xn → a as n → 00, for all fixed 2 <p < 3? FYI: Example 5.2.6 In order to appreciate the importance of Khinchine's WLLN (Theorem 5.2.3). let us consider a sequence of iid random...
4. Compute each probability or quantile 1 (а) X~ N(3, 0.0225), P(X < 3.25). (b) X~...
4. Compute each probability or quantile 1 (а) X~ N(3, 0.0225), P(X < 3.25). (b) X~ N(52, 49), Р(X > 60). (с) X~ N(3.7, 4.55), Р(3.0 < х < 10.0). (d) XN(25, 36). Find the first and third quartiles for X.
1. Suppose that X, X, X, are iid Berwulli(p),0 <p<1. Let U. - x Show that, U, can be approximated by the N (np, np(1-P) distribution, for large n and fixed <p<1. 2. Suppose that X1, X3, X. are iid N ( 0°). Where and a both assumed to be unknown. Let @ -( a). Find jointly sufficient statistics for .
Let X1,X2 be two independent
exponential random variables with λ=1, compute the
P(X1+X2<t) using the joint density function. And let Z be gamma
random variable with parameters (2,1). Compute the probability that
P(Z < t). And what you can find by comparing P(X1+X2<t) and
P(Z < t)? And compare P(X1+X2+X3<t) Xi iid
(independent and identically distributed) ~Exp(1) and P(Z < t)
Z~Gamma(3,1) (You don’t have to compute)
(Hint: You can use the fact that Γ(2)=1,
Γ(3)=2)
Problem 2[10 points] Let...
(4) Suppose that {X;}-1 iid random variables from a Binomial distribution Bin(m, p). Using your answer in (3) obtain an approximate 99% confidence interval for the pa- rameter p based on the MLE. Explain how you would estimate the Fisher information matrix.
5.2.5 (Example 5.2.6 Continued) Suppose thatXY are iid having the following common distribution. PC,-1-cip.i-1. 2. 3. and 2 <p < 3 Here. c c(p) (> 0) is such that Σ | P(X,-i) = 1.. Is there a real number a = a(p) such that Xn → a as n → 00, for all fixed 2 <p < 3? FYI: Example 5.2.6 In order to appreciate the importance of Khinchine's WLLN (Theorem 5.2.3). let us consider a sequence of iid random...
4. Compute each probability or quantile 1 (а) X~ N(3, 0.0225), P(X < 3.25). (b) X~ N(52, 49), Р(X > 60). (с) X~ N(3.7, 4.55), Р(3.0 < х < 10.0). (d) XN(25, 36). Find the first and third quartiles for X.
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