Please see attached **Title: Analyzing Childbirth Patterns Using Probability Distributions****Introduction**A sociologist is researching the changing patterns in the number of children that women have in a specific country. Using statistical methods, she represents the current number of children by the random variable $ X $. The associated probability distribution is explored to understand these patterns.**Probability Distribution Table**The table below illustrates the probability distribution of $ X $, the number of children per woman:| $ X $ | 0 | 1 | 2 | 3 | 4 | 5+ ||---------|-----|-----|-----|-----|-----|-----|| $ P(X) $ | 0.09 | 0.22 | $ a $ | 0.19 | 0.08 | 0 |**Questions and Exercises**(a) **Find the value of $ a $.**To find $ a $, use the property that the sum of all probabilities in a distribution equals 1.(b) **Calculate the probability that a woman has at least 2 children.**This requires finding the combined probability of having 2, 3, 4, or more children.(c) **Determine the mean of the probability distribution.**The mean of a probability distribution is calculated by summing the product of each value $ x_i $ and its probability $ P(x_i) $.---**Note:** To solve these questions, one must utilize principles of probability and statistical analysis to interpret data effectively, providing a comprehensive understanding of childbirth patterns in the given context.

(a) The value of a is, The probability distribution table is,

A sociologist is investigating the changing pattern of the number of children which wpmen have in a country. She denotes the present number by the random variable X which she finds to have the following probability distribution. x0 1 2| 3 4 5+ |P(X) 0.09 0.22 | a 0.19 0.080 (a) Find the value of a. (a) (b) Find the probability that a woman has at least 2 children. (b) (c) Find the mean of the probability distribution. (c)

A sociologist is investigating the changing pattern of the number of children which wpmen have in a country. She denotes the present number by the random variable X which she finds to have the following probability distribution. x0 1 2| 3 4 5+ |P(X) 0.09 0.22 | a 0.19 0.080 (a) Find the value of a. (a) (b) Find the probability that a woman has at least 2 children. (b) (c) Find the mean of the probability distribution. (c)

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Concept explainers

Estimation

A new framework of data analysis where the new statistical methods are used for interpreting the results and analyzing the data is known as estimation in statistics.

Question

Please see attached

**Title: Analyzing Childbirth Patterns Using Probability Distributions**

**Introduction**

A sociologist is researching the changing patterns in the number of children that women have in a specific country. Using statistical methods, she represents the current number of children by the random variable $ X $. The associated probability distribution is explored to understand these patterns.

**Probability Distribution Table**

The table below illustrates the probability distribution of $ X $, the number of children per woman:

| $ X $ | 0 | 1 | 2 | 3 | 4 | 5+ |
|---------|-----|-----|-----|-----|-----|-----|
| $ P(X) $ | 0.09 | 0.22 | $ a $ | 0.19 | 0.08 | 0 |

**Questions and Exercises**

(a) **Find the value of $ a $.**

To find $ a $, use the property that the sum of all probabilities in a distribution equals 1.

(b) **Calculate the probability that a woman has at least 2 children.**

This requires finding the combined probability of having 2, 3, 4, or more children.

(c) **Determine the mean of the probability distribution.**

The mean of a probability distribution is calculated by summing the product of each value $ x_i $ and its probability $ P(x_i) $.

---

**Note:** To solve these questions, one must utilize principles of probability and statistical analysis to interpret data effectively, providing a comprehensive understanding of childbirth patterns in the given context.

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