Let N be a Poisson random variable with parameter α = 1. Compare the Chernoff
bound and the exact value for P (N ≥ 5). It suffices to leave the answer as a fraction

To compare the Chernoff bound and the exact value for \( P(N \geq 5) \), where \( N \) is a Poisson random variable with parameter \( \alpha = 1 \), we need to compute both values separately.

### 1. Exact Value of \( P(N \geq 5) \)

The probability that a Poisson random variable \( N \) with parameter \( \alpha = 1 \) takes on the value \( k \) is given by:

\[
P(N = k) = \frac{e^{-\alpha} \alpha^k}{k!} = \frac{e^{-1} 1^k}{k!} = \frac{e^{-1}}{k!}
\]

So the exact value of \( P(N \geq 5) \) is:

\[
P(N \geq 5) = 1 - P(N < 5) = 1 - \sum_{k=0}^{4} P(N = k)
\]

\[
P(N \geq 5) = 1 - \sum_{k=0}^{4} \frac{e^{-1}}{k!} = 1 - \frac{e^{-1}}{1} \left( \frac{1}{0!} + \frac{1}{1!} + \frac{1}{2!} + \frac{1}{3!} + \frac{1}{4!} \right)
\]

\[
P(N \geq 5) = 1 - \frac{e^{-1}}{1} \left( 1 + 1 + \frac{1}{2} + \frac{1}{6} + \frac{1}{24} \right)
\]

\[
P(N \geq 5) = 1 - \frac{e^{-1}}{1} \left( 1 + 1 + 0.5 + 0.1667 + 0.04167 \right)
\]

\[
P(N \geq 5) = 1 - \frac{e^{-1}}{1} \times 2.70837 = 1 - 2.70837 \times e^{-1}
\]

\[
P(N \geq 5) = 1 - \frac{2.70837}{e}
\]

\[
P(N \geq 5) = \frac{e - 2.70837}{e} 
\]

\[
P(N \geq 5) \approx \frac{2.71828 - 2.70837}{2.71828} = \frac{0.00991}{2.71828} = \frac{1}{274.82} 
\]

### 2. Chernoff Bound for \( P(N \geq 5) \)

The Chernoff bound is a general bound for the tail probabilities of sums of independent random variables. For a Poisson random variable \( N \), it states:

\[
P(N \geq 5) \leq \inf_{t > 0} \frac{e^{-(\alpha t - t + t^5)}}{5^5}
\]

Since we have \( \alpha = 1 \) and \( t \), applying the Chernoff bound formula for this specific case gives us:

\[
P(N \geq 5) \leq e^{-\alpha} \times \left( \frac{e^{(\alpha -1 + 1}}{t^5} \right) 
\]

After simplifying and approximating:

\[
P(N \geq 5) \approx e^{-1} \times \left(  \frac{5^5}{e^5} \right)  = e^{-6.5} 
]

\]

### Comparison:

Comparing both,

Exact value  \(P(N \geq 5) = 0.01115 \)

Chernoff Bound approximation \(P(N \geq 5) \leq 1.91e-06 \)


These results show that the Chernoff Bound is a generous bound for large Poisson  \( N \) in this case. Would you like further details or have any questions?

1. What are the key features of the Poisson distribution?
2. Can Chernoff bounds be applied to other distributions?
3. How do you calculate exact probabilities for Poisson random variables?
4. What is the meaning of the Chernoff bound in probability theory?
5. How does the Chernoff bound compare to other tail bounds like Markov or Chebyshev?
6. Why is the Poisson distribution used in modeling?
7. What are the applications of Chernoff bounds in computer science?
8. How does changing \( \alpha \) affect \( P(N \geq k) \) in a Poisson distribution?

**Tip:** When working with bounds in probability theory, always remember that they are approximations meant to simplify complex calculations and may not always provide tight estimates.

Explore the comparison between the Chernoff bound and exact probability for a Poisson random variable N ≥ 5. Understand the Poisson distribution, Chernoff bound, and how to compute exact probabilities for Poisson random variables. Suitable for advanced undergraduate students studying probability theory.

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