Let X ∼ Normal(0, σ^2 ).
(a) Find the distribution of X^2/σ^2 . (Hint: It is a pivot
quantity.)
(b) Give an interval (L, U), where U and L are based on X, such
that P(L < σ^2 < U) = 0.95.
(c) Give an upper bound U based on X such that P(σ^2 < U) =
0.95.
(d) Give a lower bound L based on X such that P(L < σ^2 ) =
0.95
Let Z ∼ Normal(0 , 1) and Y ∼ χ 2 γ , then the new r.v. T = √ Z
Y /γ has the Student’s t-distribution. The density function of T is
fT (t) = Γ[(γ + 1)/2] √γπΓ(γ/2) 1 + t 2 γ !−(γ+1)/2 .
(a) (3 points) Describe the similarity/difference between T and
Z.
(b) (6 points) Let t0 be a particular value of t. Use
t-distribution table to find t0 values such that the following
statements...
Let X, Y ⊂ Z and x, y ∈ Z Let A = (X\{x}) ∪ {x}.
a) Prove or disprove: A ⊆ X
b) Prove or disprove: X ⊆ A
c) Prove or disprove: P(X ∪ Y ) ⊆ P(X) ∪ P(Y ) ∪ P(X ∩ Y )
d) Prove or disprove: P(X) ∪ P(Y ) ∪ P(X ∩ Y ) ⊆ P(X ∪ Y )
1A) Use surface integral to evaluate the flux
of
F(x,y,z) =<x^3,y^3,z^3>
across the cylinder x^2+y^2=1, 0<=z<=2
1B) Use the Divergence Theorem to evaluate the
flux of F(x,y,z) =<x^3,y^3,z^3>
across the cylinder x^2+y^2=1, 0<=z<=2
Mystery(y, z: positive integer)
1 x=0
2 while z > 0
3 if z mod 2 ==1 then
4
x = x + y
5 y = 2y
6 z =
floor(z/2)
//floor is the rounding down operation
7 return x
Simulate this algorithm for y=4 and z=7 and answer the following
questions:
(3 points) At the end of the first execution of the while loop,
x=_____, y=______ and z=_______.
(3 points) At the end of the second execution of...
Let z denote a random variable having a normal
distribution with μ = 0 and σ = 1. Determine each of the
probabilities below. (Round all answers to four decimal
places.)
(a) P(z < 0.2) =
(b) P(z < -0.2) =
(c) P(0.40 < z < 0.86) =
(d) P(-0.86 < z < -0.40)
=
(e) P(-0.40 < z < 0.86) =
(f) P(z > -1.24) =
(g) P(z < -1.5 or z > 2.50) =
Let z denote a random variable having a normal
distribution with μ = 0 and σ = 1. Determine each
of the following probabilities. (Round all answers to four decimal
places.)
(a)
P(z < 0.1) =
(b)
P(z < −0.1) =
(c)
P(0.40 < z < 0.85)
=
(d)
P(−0.85 < z < −0.40)
=
(e)
P(−0.40 < z < 0.85)
=
(f)
P(z > −1.25) =
(g)
P(z < −1.5 or
z > 2.50) =
Let z denote a...