The random variable X has the Poisson distribution with parameter λ > 0. According
to Chebeyshev’s Theorem, P(s < X < 99) ≥ 3/4.
1. Determine the value of k in the Chebeyshev’s inequality.
2. Determine the value of s
Consider a binomial random variable X with parameters n = 100 and p = 0.1, and
let Y be a Poisson random variable with E(Y ) = 2. Suppose that V ar(X+3Y ) = 21.
1. Find E(X), V (X) and V (Y ).
2. Determine E(X + 3Y ) and deduce E[(X + 3Y )
2
].
3. Find Cov(X, Y ). Are X and Y independent ? Justify
1. In Chebyshev's inequality, the value of k is the number of standard deviations away from the mean that we want to bound the probability. It is calculated as the square root of the reciprocal of the desired probability.
In this case, the desired probability is P(s < X < 99) ≥ 3/4. To bound the probability, we want to find the value of k such that P(|X - μ| < kσ) ≥ 3/4, where σ is the standard deviation of the Poisson distribution.
The standard deviation of a Poisson distribution is given by the square root of the parameter λ. So, σ = √λ.
Using the Chebyshev inequality:
P(|X - λ| < k√λ) ≥ 1 - (1/k^2)
We want to find the value of k such that 1 - (1/k^2) ≥ 3/4.
Solving the inequality:
1/k^2 ≤ 1/4
k^2 ≥ 4
Since k must be positive, we take the square root of both sides:
k ≥ 2
Therefore, the value of k in Chebyshev's inequality is 2.
2. To determine the value of s in the inequality, we need to find the mean of the Poisson distribution.
The mean of a Poisson distribution is equal to its parameter, so μ = λ.
Since we want to find P(s < X < 99), we need to find the lower bound s.
Using Chebyshev's inequality:
P(|X - μ| < kσ) ≥ 3/4
We substitute k = 2 and σ = √λ:
P(|X - λ| < 2√λ) ≥ 3/4
We want to find the lower bound, s, so we set |X - λ| = s:
|X - λ| = s
s = 2√λ
Therefore, the value of s in the inequality is 2√λ.
To determine the value of k in Chebyshev's inequality, we need to use the formula:
P(|X - µ| ≥ kσ) ≤ 1/k^2
Here, X is the random variable with a Poisson distribution, µ is the mean of X (which is equal to λ for a Poisson distribution), and σ is the standard deviation of X.
1. Finding the value of k:
From Chebyshev's inequality, we have P(s < X < 99) ≥ 3/4. Let's replace s with µ - kσ and solve for k.
P(µ - kσ < X < 99) ≥ 3/4
P(|X - µ| < 99 - µ + kσ) ≥ 3/4
Using Chebyshev's inequality, we can rewrite the inequality as:
1 - P(|X - µ| ≥ 99 - µ + kσ) ≤ 1 - 3/4
P(|X - µ| ≥ 99 - µ + kσ) ≤ 1/4
Since a Poisson distribution has a mean equal to its standard deviation (µ = σ), we can simplify further:
P(|X - µ| ≥ 99 - µ + kµ) ≤ 1/4
P(|X - µ| ≥ (100 - k)µ) ≤ 1/4
Now, we know from the formula that P(|X - µ| ≥ kσ) ≤ 1/k^2. So we have:
(100 - k)µ = kσ
Simplifying, we get:
(100 - k)λ = k√λ
Solving this equation will give us the value of k:
100λ - kλ = k√λ
100 - k = √λ
k = 100 - √λ
Therefore, the value of k is 100 - √λ.
2. Determining the value of s:
From the equation s < X < 99, we can substitute s with µ - kσ:
µ - kσ < X < 99
Since µ = λ and σ = √λ for a Poisson distribution, we can rewrite it as:
λ - (100 - √λ)√λ < X < 99
Solving for λ, we can find the value of s:
λ - (100 - √λ)√λ = s
λ - (100√λ - λ) = s
s = 100√λ - λ
Therefore, the value of s is 100√λ - λ.