Math 5090
Quiz 1
Name:
Show all work clearly and in order, and circle or box your final answers. Justify your answers
mathematically whenever possible; when you do use your calculator, sketch all relevant graphs
and write down all relevant mathematics.
1.) Let X1 , X2 , . . . , Xn denote an i.i.d. sequence of random variables with density
h(t, µ, σ) = √
1
2πσ 2
e−(t−µ)
2 /2σ 2
.
√
√
We use [X̄n − σ/ n, X̄n + 3σ/2 n] as a confidence interval for µ. Compute the probability that
µ is in this interval.
2.) Let X1 , X2 , . . . , Xn denote an i.i.d. sequence of random variables with density
3 t 2
f (t, θ) =
I(0 < t < θ).
θ θ
Find a 100(1 − α)% equal-tail confidence interval for θ based on a pivotal variable.
3.) Let X1 , X2 , . . . , Xn denote an i.i.d. sequence of random variables with density
f (t, θ) = e−(t−θ) I(θ ≤ t < ∞).
Find constants c1 and c2 such that [X1,n − c1 , X1,n + c2 ] is a 100(1 − α)% equal-tail confidence
interval for θ.