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?


Let's solve each part of the problem step by step, given that \( \lambda = 5 \) and the exponential distribution with the cumulative distribution function (CDF):

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

### Part (a): Probability that \( X \in (0, 6) \)

To find \( P(0 < X < 6) \), we use the cumulative distribution function at \( z = 6 \):

\[
P(X < 6) = 1 - e^{-5 \cdot 6} = 1 - e^{-30}
\]

Since we know that \( P(0 < X < 6) = P(X < 6) \), the result is:

\[
P(0 < X < 6) = 1 - e^{-30}
\]

Since \( e^{-30} \) is a very small number, the result will be almost 1:

\[
P(0 < X < 6) \approx 1
\]

### Part (b): Probability that \( X \in (2, 6) \)

To calculate \( P(2 < X < 6) \), we compute the difference in probabilities:

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

Using the CDF at \( z = 6 \) and \( z = 2 \):

\[
P(X < 6) = 1 - e^{-5 \cdot 6} = 1 - e^{-30} \approx 1
\]
\[
P(X < 2) = 1 - e^{-5 \cdot 2} = 1 - e^{-10}
\]

Thus:

\[
P(2 < X < 6) = (1 - e^{-30}) - (1 - e^{-10}) = e^{-10} - e^{-30}
\]

Since \( e^{-30} \) is negligibly small, the result is approximately:

\[
P(2 < X < 6) \approx e^{-10} = 4.53999 \times 10^{-5}
\]

### Part (c): Probability that at least one \( X_i \) is greater than 1

We are asked to find the probability that at least one of the 7 samples \( X_1, X_2, \ldots, X_7 \) is greater than 1. First, let's find the probability that each individual \( X_i \) is less than 1:

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

Thus, the probability that each \( X_i \) is greater than 1 is:

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

Now, the probability that all 7 samples are less than or equal to 1 is:

\[
P(\text{all } X_i \leq 1) = (1 - e^{-5})^7
\]

Finally, the probability that at least one \( X_i \) is greater than 1 is:

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

Let me calculate the values for these expressions.Here are the results for each part:

### Part (a):
The probability that \( X \in (0, 6) \) is approximately:
\[
P(0 < X < 6) \approx 1
\]

### Part (b):
The probability that \( X \in (2, 6) \) is approximately:
\[
P(2 < X < 6) \approx 4.54 \times 10^{-5}
\]

### Part (c):
The probability that at least one of the 7 samples is greater than 1 is approximately:
\[
P(\text{at least one } X_i > 1) \approx 0.0462
\]

Would you like more details on any of these calculations?

Here are 5 related questions to expand your understanding:
1. How is the CDF of the exponential distribution derived from its PDF?
2. What is the significance of the parameter \( \lambda \) in the exponential distribution?
3. How would the probabilities change if \( \lambda \) were smaller, such as \( \lambda = 2 \)?
4. How can we calculate the mean and variance of an exponential distribution?
5. What is the relationship between exponential and Poisson distributions?

**Tip:** The exponential distribution is often used to model the time between events in a Poisson process, where events occur continuously and independently.

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?

Solve an exponential distribution problem where λ = 5. Learn how to calculate the probabilities for X ∈ (0, 6), X ∈ (2, 6), and the probability that at least one of 7 samples is greater than 1. Detailed step-by-step solution and explanation of key concepts.

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