3.	The exponential distribution is a commonly used continuous distribution, with a density function defined by:
	p(x) = λ e−λx,		x ∈ [0, ∞)
Where λ > 0.  It can be shown (using calculus) that the area under the curve over the interval [0, z], and hence the probability that P (X < z), is given by:
P (X < z) = 1 − e−λz
Suppose λ = 5,
a)	What is the probability that X ∈ (0, 6)?
b)	What is the probability that X ∈ (2, 6)?
c)	Suppose 7 samples X1, X2, . . ., X7 are sampled from this distribution, meaning
 P (Xi < z) = 1 − e−5z, what is the probability that at least one of the Xi  is greater than 1?


We are working with the exponential distribution where the probability density function (PDF) is defined as:

\[
p(x) = \lambda e^{-\lambda x}, \quad x \in [0, \infty)
\]

The cumulative distribution function (CDF), which gives the probability \( P(X < z) \), is:

\[
P(X < z) = 1 - e^{-\lambda z}
\]

Given \( \lambda = 5 \), we are tasked with calculating the following probabilities:

### a) What is the probability that \( X \in (0, 6) \)?

The probability that \( X \in (0, 6) \) is the same as \( P(X < 6) \), because the distribution starts at \( x = 0 \). Thus:

\[
P(X < 6) = 1 - e^{-5 \cdot 6}
\]

Let's calculate that.

\[
P(X < 6) = 1 - e^{-30}
\]

Since \( e^{-30} \) is a very small number, we have:

\[
P(X < 6) \approx 1
\]

### b) What is the probability that \( X \in (2, 6) \)?

The probability that \( X \in (2, 6) \) is calculated as:

\[
P(2 < X < 6) = P(X < 6) - P(X < 2)
\]

Using the CDF formula:

\[
P(2 < X < 6) = \left( 1 - e^{-5 \cdot 6} \right) - \left( 1 - e^{-5 \cdot 2} \right)
\]

This simplifies to:

\[
P(2 < X < 6) = e^{-5 \cdot 2} - e^{-5 \cdot 6}
\]

Let's compute that:

\[
P(2 < X < 6) = e^{-10} - e^{-30}
\]

Since \( e^{-30} \) is very small, we approximate:

\[
P(2 < X < 6) \approx e^{-10}
\]

### c) What is the probability that at least one of the \( X_i \) is greater than 1, given that 7 samples \( X_1, X_2, \dots, X_7 \) are taken from this distribution?

The probability that a single sample \( X_i \) is less than 1 is:

\[
P(X_i < 1) = 1 - e^{-5 \cdot 1} = 1 - e^{-5}
\]

Therefore, the probability that all 7 samples are less than 1 is:

\[
P(\text{All } X_i < 1) = \left( 1 - e^{-5} \right)^7
\]

The probability that **at least one** of the \( X_i \)'s is greater than 1 is the complement:

\[
P(\text{At least one } X_i > 1) = 1 - \left( 1 - e^{-5} \right)^7
\]

Let's compute that using \( e^{-5} \approx 0.0067 \):

\[
P(\text{At least one } X_i > 1) = 1 - (1 - 0.0067)^7
\]

Now, we can calculate the final value.

---

Would you like further details or have any questions?

Here are 5 related questions for further exploration:

1. What is the expected value and variance of the exponential distribution?
2. How would the probabilities change if \( \lambda \) were a different value?
3. How is the exponential distribution related to the Poisson process?
4. Can you calculate the probability that \( X \in (4, 8) \) for this distribution?
5. What is the median of the exponential distribution?

**Tip**: The exponential distribution is often used to model waiting times in processes where events occur continuously and independently at a constant rate (e.g., time between phone calls).

The exponential distribution is a commonly used continuous distribution, with a density function defined by: p(x) = λ e−λx, x ∈ [0, ∞) Where λ > 0. It can be shown (using calculus) that the area under the curve over the interval [0, z], and hence the probability that P(X < z), is given by: P(X < z) = 1 − e−λz Suppose λ = 5, a) What is the probability that X ∈ (0, 6)? b) What is the probability that X ∈ (2, 6)? c) Suppose 7 samples X1, X2, . . ., X7 are sampled from this distribution, meaning P(Xi < z) = 1 − e−5z, what is the probability that at least one of the Xi is greater than 1?

Learn how to calculate probabilities for the exponential distribution with λ = 5. This example covers calculating the probability that X lies in intervals such as (0, 6) and (2, 6), and finding the probability that at least one of seven samples is greater than 1. Suitable for undergraduate students studying probability theory.

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