ramj

Let X1,…,Xn be i.i.d. Poisson random variables with parameter λ>0 and denote by X¯¯¯¯n their empirical average, X¯¯¯¯n=1n∑i=1nXi. Find two sequences (an)n≥1 and (bn)n≥1 such that an(X¯¯¯¯n−bn) converges in distribution to a standard Gaussian random variab...

Let 𝑋1,…,𝑋𝑛 be i.i.d. random variables with distribution (𝜃,𝜃) , for some unknown parameter 𝜃>0 . Find an interval I𝜃 (that depends on 𝜃 ) centered about 𝑋⎯⎯⎯⎯⎯𝑛 such that P(I𝜃 <- 𝜃 )=0.9for all 𝑛(i.e, not only for large 𝑛). (Write barX_n for 𝑋⎯⎯⎯⎯⎯𝑛 . U...

The Gamma distribution Gamma(𝛼,𝛽) with paramters 𝛼>0 , and 𝛽>0 is defined by the density 𝑓𝛼,𝛽(𝑥)=𝛽𝛼Γ(𝛼)𝑥𝛼−1𝑒−𝛽𝑥,for all𝑥≥0. The Γ function is defined by Γ(𝑠)=∫∞0𝑥𝑠−1𝑒−𝑥𝑑𝑥. As usual, the constant 𝛽𝛼Γ(𝛼) is a normalization constant that gives ∫∞0𝑓𝛼,𝛽(𝑥)𝑑𝑥=1...

The National Assessment of Educational Progress tested a simple random sample of 1000 thirteen year old students in both 2004 and 2008 and recorded each student's score. The average and standard deviation in 2004 were 257 and 39, respectively. In 2008, th...

a)𝑋1,…,𝑋𝑛∼𝑖.𝑖.𝑑.𝖯𝗈𝗂𝗌𝗌(𝜆) for some unknown 𝜆>0 ; 𝐻0:𝜆=𝜆0 v.s. 𝐻1:𝜆≠𝜆0where 𝜆0>0. (Type barX_n for 𝑋⎯⎯⎯⎯⎯𝑛 , lambda_0 for 𝜆0 . If applicable, type abs(x) for |𝑥| , Phi(x) for Φ(𝑥)=𝐏(𝑍≤𝑥) where 𝑍∼N(0,1) , and q(alpha) for 𝑞𝛼 , the 1−𝛼 quantile of a standard...

Let 𝑋1,…,𝑋𝑛 be i.i.d. Bernoulli random variables with unknown parameter 𝑝∈(0,1) . Suppose we want to test 𝐻0:𝑝∈[0.48,0.51]vs𝐻1:𝑝∉[0.48,0.51] We want to construct an asymptotic test 𝜓 for these hypotheses using 𝑋⎯⎯⎯⎯⎯𝑛. For this problem, we specifically co...

Let X be a single (i.e. n=1 ) Gaussian random variable with unknown mean μ and variance 1 . Consider the following hypotheses: H0:μ=0vsH1:μ≠0. Define a test ψα:R→{0,1} with level α that is of the form ψα=1{fα(X)>0}, for some function fα:R→R . We want our...

Suppose that the rejection region of a test ψ has the form R={Xn:Xn>c} . Find the smallest c such that ψ has level α . (If applicable, type abs(x) for |x| , Phi(x) for Φ(x)=P(Z≤x) where Z∼N(0,1) , and q(alpha) for qα , the 1−α quantile of a standard norma...

Secondly, express P(|Z|≤t) in terms of Φ(r)=P(Z≤r) for t>0 . Write Phi(t) (with capital P) for Φ(t) .

clarification : Let 𝑋1,…,𝑋𝑛 be i.i.d. random variables with distribution N(𝜃,𝜃) , for some unknown parameter 𝜃>0 .

Using the previous part, find confidence intervals for 𝛼 with asymptotic level 90% using both the “solving" and the “plug-in" methods. Use 𝑛=25 , and 𝑋⎯⎯⎯⎯⎯𝑛=4.5 . (Enter your answers accurate to 2 decimal places. Use the Gaussian estimate 𝑞0.05≈1.6448 fo...

Suppose that the test ψ has level α=0.05 . What is the power of ψ ? (If applicable, type abs(x) for |x| , Phi(x) for Φ(x)=P(Z≤x) where Z∼N(0,1) , and q(alpha) for qα , the 1−α quantile of a standard normal variable, e.g. enter q(0.01) for q0.01 .) Power o...