Question

For all problems on this page, use the following setup:

Let N be a positive integer random variable with PMF of the form

pN(n)=1/2⋅n⋅2^(−n),n=1,2,…. |

Once we see the numerical value of N, we then draw a random variable K whose (conditional) PMF is uniform on the set {1,2,…,2n}.

Write down an expression for the joint PMF pN,K(n,k).

For n=1,2,… and k=1,2,…,2n:

pN,K(n,k)=

Answer #1

TOPIC:Joint pmf from the conditional pmf and the marginal pmf.

Let N be a positive integer random variable with PMF of the form
pN(n)=12⋅n⋅2−n,n=1,2,…. Once we see the numerical value of N , we
then draw a random variable K whose (conditional) PMF is uniform on
the set {1,2,…,2n} . 1. Find joint PMF pN,K(n,k) For n=1,2,… and
k=1,2,…,2n 2. Find the marginal PMF pK(k) as a function of k . For
simplicity, provide the answer only for the case when k is an even
number. For k=2,4,6,… 3. Let...

Let N be a positive integer random variable with PMF of the
form
pN(n)=1/2⋅n⋅2^(−n),n=1,2,….
Once we see the numerical value of N, we then draw a random
variable K whose (conditional) PMF is uniform on the set
{1,2,…,2n}.
Find the marginal PMF pK(k) as a function of k. For simplicity,
provide the answer only for the case when k
is an even number. (The formula for when k is odd
would be slightly different, and you do not need to...

Let K be a random variable that takes, with equal probability
1/(2n+1), the integer values in the interval [-n,n].
Find the PMF of the random variable Y = In X. Where X = a^[k]. and
a is a positive number, let n = 7 and a = 2. Then what is E[Y
]?

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 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 f(n) be a negligible function and k a positive integer.
Prove the following:
(a) f(√n) is negligible.
(b) f(n/k) is negligible.
(c) f(n^(1/k)) is negligible.

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)....

Euler's Totient Function
Let f(n) denote Euler's totient function; thus, for a positive
integer n, f(n) is the number of integers less than n which are
coprime to n. For a prime p its is known that f(p^k) = p^k-p^{k-1}.
For example f(27) = f(3^3) = 3^3 - 3^2 = (3^2) 2=18. In addition,
it is known that f(n) is multiplicative in the sense that
f(ab) = f(a)f(b)
whenever a and b are coprime. Lastly, one has the celebrated
generalization...

1.A fair die is rolled once, and the number score is noted.
Let the random variable X be twice this score. Define the variable
Y to be zero if an odd number appears and X otherwise. By finding
the probability mass function in each case, find the expectation of
the following random variables:
Please answer to 3 decimal places.
Part a)X
Part b)Y
Part c)X+Y
Part d)XY
——-
2.To examine the effectiveness of its four annual advertising
promotions, a mail...

Please read the article and answear about
questions.
Determining the Value of the Business
After you have completed a thorough and exacting investigation,
you need to analyze all the infor- mation you have gathered. This
is the time to consult with your business, financial, and legal
advis- ers to arrive at an estimate of the value of the business.
Outside advisers are impartial and are more likely to see the bad
things about the business than are you. You should...

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