Question

Show that if x ∈ P, then x^(-1) ∈ P. Hint: show that a
contradiction will follow if one assumes that x ∈ P and x ∉
P.

Answer #1

Let p and q be two real numbers with p > 0. Show that the
equation x^3 + px +q= 0 has exactly one real solution.
(Hint: Show that f'(x) is not 0 for any real x and then use
Rolle's theorem to prove the statement by contradiction)

If X∼Binom(n,p), E[X] = np. Calculate V[X] = np(1−p) by:
a) First show E[X(X−1)] + E[X] − (E[X])^2 = V[X] (Hint: Use
propertie of E[·] and V[·]).
b) Show E[X(X−1)] = n(n−1)p^2
c) Use E[X] = np, a) and b) to discuss, V[X] = np(1−p).

Prove by contradiction that 5√ 2 is an irrational number. (Hint:
Dividing a rational number by another rational number yields a
rational number.)

: Prove by contradiction that 5√ 2 is an irrational number.
(Hint: Dividing a rational number by another rational number yields
a rational number.)

For two events A and B show that P (A∩B) ≥ P (A)+P (B)−1. (Hint:
Apply de Morgan’s law and then the Bonferroni inequality). Derive
below Results 1 to 4 from Axioms 1 to 3 given in Section 2.1.2 in
the textbook.
Result 1: P (Ac) = 1 − P(A)
Result 2 : For any two events A and B, P (A∪B) = P (A)+P (B)−P
(A∩B)
Result 3: For any two events A and B, P(A) = P(A ∩...

Xand Y are independent X~G(p) , Y~G(p)
show U=X+Y follow NB(2,p)

. Consider the Bernoulli distribution, P(X = x|p) = (p^x) (1 −
p) ^(1−x) for x = 0 and x = 1.
(a) Show that this is an exponential family.
(b) Find a sufficient statistic for p.
(c) Show that X is a m.v.u.e. for p.

Which of the following are valid probability distribution
functions? (Hint: The P values have to follow the rules previously
given for probabilities.)
x
P(x)
5
0.6
6
0.8
7
– 0.4
x
P(x)
1
.1
2
.2
3
.3
4
.4
x
0
1
2
3
4
P(x)
1/6
1/6
1/6
1/6
1/6

Let X = (xn) be a sequence in R^p which is convergent
to x. Show that lim(||xn||) = ||x||. hint: use triange
inequality

Show if X ~ F( p, q) , then [(p/q) X]/[1+(p/q)X] ~ beta (p/2,
q/2). Use transformation method.

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