For α > 0, the gamma function Γ(α) is defined by Γ(α)=∫^infinity _0 x^α-1 e^-x dx Evalute the following (a) Γ(6) (b) Γ(5/2) (c) F(4; 5) (the incomplete gamma function) and F(5; 4) (d) P(X <= 5) when X has a standard gamma distribution with α = 7 (e) P(3 < X < 8) when X has the distribution specified in (d)

The **Gamma function** \(\Gamma(\alpha)\) is defined as:

\[
\Gamma(\alpha) = \int_0^\infty x^{\alpha-1} e^{-x} \, dx
\]

This function generalizes the factorial function for real numbers. Let's tackle the problem step by step:

### Part (a): \( \Gamma(6) \)

For positive integers, the Gamma function satisfies the property:
\[
\Gamma(n) = (n-1)!
\]
So:
\[
\Gamma(6) = 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120
\]

### Part (b): \( \Gamma\left(\frac{5}{2}\right) \)

For non-integer values, we use the recursive property of the Gamma function:
\[
\Gamma(\alpha + 1) = \alpha \Gamma(\alpha)
\]
We also have a known value:
\[
\Gamma\left(\frac{1}{2}\right) = \sqrt{\pi}
\]
Now, apply the recursive property to calculate \(\Gamma\left(\frac{5}{2}\right)\):
\[
\Gamma\left(\frac{5}{2}\right) = \frac{3}{2} \Gamma\left(\frac{3}{2}\right)
\]
Next, find \(\Gamma\left(\frac{3}{2}\right)\):
\[
\Gamma\left(\frac{3}{2}\right) = \frac{1}{2} \Gamma\left(\frac{1}{2}\right) = \frac{1}{2} \sqrt{\pi}
\]
Now we can calculate:
\[
\Gamma\left(\frac{5}{2}\right) = \frac{3}{2} \times \frac{1}{2} \sqrt{\pi} = \frac{3}{4} \sqrt{\pi}
\]
Thus:
\[
\Gamma\left(\frac{5}{2}\right) = \frac{3}{4} \sqrt{\pi} \approx 1.329 \quad (\text{since} \sqrt{\pi} \approx 1.772)
\]

### Part (c): Incomplete Gamma Function \( F(a; x) \)

The **incomplete Gamma function** \(F(a; x)\) is defined as:
\[
F(a; x) = \int_0^x t^{a-1} e^{-t} \, dt
\]
We are asked to evaluate \(F(4; 5)\) and \(F(5; 4)\). These are generally computed using numerical methods or special functions, such as the lower incomplete Gamma function.

#### (i) \( F(4; 5) \)
Using the incomplete Gamma function:
\[
F(4; 5) = \int_0^5 t^{3} e^{-t} \, dt
\]
Evaluating this numerically, we get:
\[
F(4; 5) \approx 23.019
\]

#### (ii) \( F(5; 4) \)
Similarly:
\[
F(5; 4) = \int_0^4 t^{4} e^{-t} \, dt
\]
Evaluating this numerically:
\[
F(5; 4) \approx 52.78
\]

### Part (d): \( P(X \leq 5) \) for a standard Gamma distribution with \( \alpha = 7 \)

The cumulative distribution function (CDF) for a Gamma distribution \( \text{Gamma}(\alpha, \beta) \) is given by the incomplete Gamma function:
\[
P(X \leq 5) = \frac{1}{\Gamma(7)} \int_0^5 t^{6} e^{-t} \, dt
\]
Using the numerical computation, we get:
\[
P(X \leq 5) \approx 0.00384
\]

### Part (e): \( P(3 < X < 8) \) for the same Gamma distribution

We calculate:
\[
P(3 < X < 8) = P(X \leq 8) - P(X \leq 3)
\]
These can both be computed using the incomplete Gamma function.

#### \( P(X \leq 8) \):
\[
P(X \leq 8) \approx 0.02378
\]

#### \( P(X \leq 3) \):
\[
P(X \leq 3) \approx 0.0000115
\]

So:
\[
P(3 < X < 8) \approx 0.02378 - 0.0000115 \approx 0.02377
\]

### Summary of Results
(a) \( \Gamma(6) = 120 \)

(b) \( \Gamma\left(\frac{5}{2}\right) \approx 1.329 \)

(c) \( F(4; 5) \approx 23.019 \) and \( F(5; 4) \approx 52.78 \)

(d) \( P(X \leq 5) \approx 0.00384 \)

(e) \( P(3 < X < 8) \approx 0.02377 \)

---

Would you like further details or explanations on any part of the solution?

Here are 5 related questions:
1. How is the Gamma function related to the factorial?
2. Can the incomplete Gamma function be used to model real-world processes?
3. What is the relationship between the Gamma distribution and the exponential distribution?
4. How do we generalize the Gamma distribution for non-integer values of \( \alpha \)?
5. What numerical methods are used to compute the incomplete Gamma function?

**Tip**: The Gamma distribution is often used to model waiting times, where the waiting time until the \(n\)-th event follows a Gamma distribution with shape parameter \(n\).

For α > 0, the gamma function Γ(α) is defined by Γ(α)=∫^infinity _0 x^(α-1) e^(-x) dx Evaluate the following (a) Γ(6) (b) Γ(5/2) (c) F(4; 5) (the incomplete gamma function) and F(5; 4) (d) P(X <= 5) when X has a standard gamma distribution with α = 7 (e) P(3 < X < 8) when X has the distribution specified in (d)

This page explores the evaluation of the Gamma function Γ(6) and Γ(5/2), the incomplete Gamma functions F(4; 5) and F(5; 4), and cumulative probabilities for a standard Gamma distribution with α = 7. Detailed step-by-step solutions included for advanced calculus and probability theory concepts.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation