Question

Let X Geom(p). For positive integers n, k define

P(X = n + k | X > n) = P(X = n + k) / P(X > n) :

Show that P(X = n + k | X > n) = P(X = k) and then briefly argue, in words, why this is true for geometric random variables.

Answer #1

Please include the argument in word, thanks
Let X ∼ Geom(p). For positive
integers n, k define
P(X = n + k | X > n) =
P(X = n + k) / P(X > n)
.
Show that P(X = n + k |
X > n) = P(X = k) and then briefly
argue, in words, why this is true
for geometric random variables.

let x be a discrete random variable with positive integer
outputs.
show that P(x=k) = P( x> k-1)- P( X>k) for any positive
integer k.
assume that for all k>=1 we have P(x>k)=q^k. use (a) to
show that x is a geometric random variable.

Let X be a discrete random
variable with positive integer outputs
a show that
p (X= K)= P( X> K-1) - P( X> k)
for any positive integer k
b Assume that for all k >I
we have P (X>k)=q^k use l()
to show that X is a geometric
random variable

Let n be a positive integer and p and
r two real numbers in the interval (0,1). Two random
variables X and Y are defined on a the same
sample space. All we know about them is that
X∼Geom(p) and
Y∼Bin(n,r). (In particular, we do not
know whether X and Y are independent.) For each
expectation below, decide whether it can be calculated with this
information, and if it can, give its value (in terms of p,
n, and r)....

Let a, b be positive integers and let a = k(a, b), b = h(a,
b).
Suppose that ab = n^2 show that k and h are perfect squares.

Let N denote the set of positive integers, and let x be a number
which does not belong to N. Give an explicit bijection f : N ∪ x →
N.

Prove that for fixed positive integers k and n, the number of
partitions of n is equal to the number of partitions of 2n + k into
n + k parts.
show by using bijection

(14pts) Let X and Y be i.i.d. geometric random variables with
parameter (probability of success) p, 0 < p < 1. (a) (6pts)
Find P(X > Y ). (b) (8pts) Find P(X + Y = n) and P(X = k∣X + Y =
n), for n = 2, 3, ..., and k = 1, 2, ..., n − 1.

3. Let N denote the nonnegative integers, and Z denote the
integers. Define the function g : N→Z defined by g(k) = k/2 for
even k and g(k) = −(k + 1)/2 for odd k. Prove that g is a
bijection.
(a) Prove that g is a function.
(b) Prove that g is an injection
. (c) Prove that g is a surjection.

1. Let n be an odd positive integer. Consider a list of n
consecutive integers.
Show that the average is the middle number (that is the number
in the
middle of the list when they are arranged in an increasing
order). What
is the average when n is an even positive integer instead?
2.
Let x1,x2,...,xn be a list of numbers, and let ¯ x be the
average of the list.Which of the following
statements must be true? There might...

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