Question

Let X~Poisson(4) random variable and Y an independent Bin(10,1/2) random variable.

(a) Use Markov's inequality to find an upper bound for P(X+Y > 15).

(b) Use Chebyshev's inequality to find an upper bound for P(X+Y > 15)

Answer #1

Let X be a Poisson random variable with parameter λ and Y an
independent Bernoulli random variable with parameter p. Find the
probability mass function of X + Y .

For a continuous random variable X , you are given that
the mean is E(X)= m and the variance is var(x)= v.
Let m=(R+L)/2
Given L = 6, R = 113 and V = 563, use Chebyshev's
inequality to compute a lower bound for the following probability
P(L<X<R)
Lower bound means that you need to find a
value such thatP(L<X<R)>p
using Chebyshev's inequality.

Let X follow Poisson distribution with λ = a and Y follow
Poisson distribution with λ = b. X and Y are independent. Define a
new random variable as Z=X+Y. Find P(Z=k).

a. Suppose X and Y are independent Poisson
random variables, each with expected value 2. Define Z=X+Y. Find
P(Z?3).
b. Consider a Poisson random variable X with
parameter ?=5.3, and its probability mass function, pX(x). Where
does pX(x) have its peak value?

Let X be a gamma random variable with parameters alpha = 4 and
beta = 4. Using Markov's inequality, calculate an upper bound for
the probability that X is greater than or equal to 10.

7.
Let X and Y be two independent and identically distributed
random variables with expected value 1 and variance 2.56.
(i) Find a non-trivial upper bound for
P(| X + Y -2 | >= 1)
(ii) Now suppose that X and Y are independent and identically
distributed N(1;2.56) random variables. What is P(|X+Y=2| >= 1)
exactly? Briefly, state your reasoning.
(iii) Why is the upper bound you obtained in Part (i) so
different from the exact probability you obtained in...

Let X and Y be independent random variables following Poisson
distributions, each with parameter λ = 1. Show that the
distribution of Z = X + Y is Poisson with parameter λ = 2. using
convolution formula

How to use Chebyshev bound to achieve this question ?
Let X be a Geometric random variable, with success probability
p.
1) Use the Markov bound to find an upper bound for P (X ≥ a), for a
positive integer a.
2) If p = 0.1, use the Chebyshev bound to find an upper bound for
P(X ≤ 1). Compare it with the
actual value of P (X ≤ 1) which you can calculate using the PMF of
Geometric random...

For a continuous random variable , you are given that
the mean is and the variance is .
Let
Given = 27, = 130 and =
592, use Chebyshev's inequality to compute a lower bound for the
following probability
Lower bound means that you need to find a
value such that
using Chebyshev's inequality.

Let X be a random variable such that P(X=k) = k/10, for k =
1,2,3,4. Let Y be a random variable with the same distribution as
X. Suppose X and Y are independent. Find P(X+Y = k), for k =
2,...,8.

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