### 3.3.2 Gamma Distribution CDF for Positive Integer $ n $If $ X \sim \text{GAM}(\theta, n) $, where $ n $ is a positive integer, then the Cumulative Distribution Function (CDF) can be written as:\[ F(x; \theta, n) = 1 - \sum_{i=0}^{n-1} \frac{(x/\theta)^i}{i!} e^{-x/\theta} \](Equation 3.3.12)In this context:- $ \theta $ (theta) is the scale parameter.- $ n $ is the shape parameter, which is a positive integer.- $ x $ is the variable of interest.- $ i $ is an index variable that iterates from 0 to $ n-1 $.The expression inside the summation is the normalized incomplete gamma function, which, together with the exponential term, represents the probability that a gamma-distributed random variable $ X \leq x $. ### Survival Time of a White Rat Under X-ray RadiationThe survival time (in days) of a white rat that was subjected to a certain level of X-ray radiation is represented as a random variable $ X $ following a Gamma distribution: $ X \sim \text{Gam}(4, 3) $. Utilize Theorem 3.3.2 to find the following:a. $ P(X < 12) $b. $ P(12 < X < 16) $c. Find $ E(X) $---**Explanation of the Gamma Distribution Parameters:**- $ k = 4 $ (shape parameter)- $ \theta = 3 $ (scale parameter)**Theorem 3.3.2** generally provides methods to calculate probabilities and expectations for Gamma-distributed random variables. Refer to your course materials or textbook for details on how to apply this theorem to solve the above problems.

Answered: Image /qna-images/answer/6761d90f-f766-4018-9052-844f94563288.jpg

Answered: The survival time (in days) of a white…

The survival time (in days) of a white rat that was subjected to a certain level of X-ray radiation is a random variable X~ Gam(4, 3). Use Theorem 3.3.2 to find: РСХ < 12) P(12

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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Concept explainers

Continuous Probability Distributions

Probability distributions are of two types, which are continuous probability distributions and discrete probability distributions. A continuous probability distribution contains an infinite number of values. For example, if time is infinite: you could count from 0 to a trillion seconds, billion seconds, so on indefinitely. A discrete probability distribution consists of only a countable set of possible values.

Normal Distribution

Suppose we had to design a bathroom weighing scale, how would we decide what should be the range of the weighing machine? Would we take the highest recorded human weight in history and use that as the upper limit for our weighing scale? This may not be a great idea as the sensitivity of the scale would get reduced if the range is too large. At the same time, if we keep the upper limit too low, it may not be usable for a large percentage of the population!

Question

### 3.3.2 Gamma Distribution CDF for Positive Integer $ n $

If $ X \sim \text{GAM}(\theta, n) $, where $ n $ is a positive integer, then the Cumulative Distribution Function (CDF) can be written as:

\[ F(x; \theta, n) = 1 - \sum_{i=0}^{n-1} \frac{(x/\theta)^i}{i!} e^{-x/\theta} \]

(Equation 3.3.12)

In this context:
- $ \theta $ (theta) is the scale parameter.
- $ n $ is the shape parameter, which is a positive integer.
- $ x $ is the variable of interest.
- $ i $ is an index variable that iterates from 0 to $ n-1 $.

The expression inside the summation is the normalized incomplete gamma function, which, together with the exponential term, represents the probability that a gamma-distributed random variable $ X \leq x $.

### Survival Time of a White Rat Under X-ray Radiation

The survival time (in days) of a white rat that was subjected to a certain level of X-ray radiation is represented as a random variable $ X $ following a Gamma distribution: $ X \sim \text{Gam}(4, 3) $. Utilize Theorem 3.3.2 to find the following:

a. $ P(X < 12) $

b. $ P(12 < X < 16) $

c. Find $ E(X) $

---

**Explanation of the Gamma Distribution Parameters:**
- $ k = 4 $ (shape parameter)
- $ \theta = 3 $ (scale parameter)

**Theorem 3.3.2** generally provides methods to calculate probabilities and expectations for Gamma-distributed random variables. Refer to your course materials or textbook for details on how to apply this theorem to solve the above problems.

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