PROBABILITY DISTRIBUTIONS

A random variable is continuous if it can take any value within an interval on the real number line.

Examples include: height, time, temperature, age, pressure, and velocity.

Key characteristics:

• The variable takes values in an interval such as \((a, b)\) or even \((-\infty, +\infty)\).

• The probability of any single point is always zero:

\[ P(X = x) = 0 \]

• Probabilities are meaningful only over intervals:

\[ P(a \le X \le b) = \int_{a}^{b} f(x)\, dx \]

A function \(f(x)\) is a valid Probability Density Function (PDF) if it satisfies:

1. Non-negativity

\[ f(x) \ge 0 \quad \forall x \]

2. Total Area Equals 1

\[ \int_{-\infty}^{\infty} f(x)\,dx = 1 \]

Interpretation:

• Larger values of \(f(x)\) indicate higher probability density around that value.

• However, \(f(x)\) is not a probability; probabilities come from the area under the curve.

Example PDF: \(f(x) = 3x^2 \text{ on } [0,1]\)

Consider the probability density function:

\[ f(x) = 3x^2,\quad 0 \le x \le 1 \]

Validation:

\[ \int_{0}^{1} 3x^2 \, dx = 1 \]

To compute probability within an interval:

\[ P(a \le X \le b) = \int_{a}^{b} 3x^2 \, dx \]

Example: \[ P(0.5 \le X \le 1) \]

The Cumulative Distribution Function (CDF) is defined as:

\[ F(x) = P(X \le x) = \int_{0}^{x} 3t^{2}\, dt = x^{3} \]

Relationship between PDF and CDF: \[ f(x) = F'(x) \]

Sample means can be seen from their distribution shape. The distribution of a sample is the distribution of a single sample taken from a population, so its shape follows the pattern of the original data—it can be right-skewed, left-skewed, or non-normal. In contrast, the sampling distribution of sample means is the distribution formed from the means of many samples of the same size, and according to the Central Limit Theorem, the distribution of these means will tend to be normal if the sample size is large enough (generally n ≥ 30), even if the original data is not normal. Therefore, if a distribution is normal in shape, it is usually the sampling distribution of sample means, whereas if the distribution is skewed, it is the distribution of a sample.

Example:

A recent study with a sample size of 200 found that the mean height of an adult in a particular city is 164 cm with a standard error of 3.7 cm and that the heights are normally distributed. Is this a description of a distribution of the sample or is it a description of the sampling distribution of sample means?

• Step 1: Analyze the information given regarding the shape of the distribution - is it normally distributed or skewed to the right or left? The data is described as being normally distributed - not skewed in either direction.

• Step 2: Use the Central Limit Theorem to conclude if the described distribution is a distribution of a sample or a sampling distribution of sample means. If the shape is skewed right or left, the distribution is a distribution of a sample. If the shape is normally distributed, the distribution is a sampling distribution of sample means. Since we have large samples (size 100) and the distribution is normal, we can conclude that the description is of a sampling distribution of sample means.

shows a side-by-side comparison of a histogram for the original population and a histogram for this distribution. Whereas the distribution of the population is uniform, the sampling distribution of the mean has a shape approaching the shape of the familiar bell curve. This phenomenon of the sampling distribution of the mean taking on a bell shape even though the population distribution is not bell-shaped happens in general. Here is a somewhat more realistic example.

Suppose we take samples of size 1, 5, 10, or 20 from a population that consists entirely of the numbers 0 and 1, half the population 0, half 1, so that the population mean is 0.5. The sampling distributions are:

n = 1:

\[ \begin{array}{c|cc} \hline \bar{x} & 0 & 1 \\ \hline P(\bar{x}) & 0.5 & 0.5 \\ \hline \end{array} \]

n = 5:

\[ \begin{array}{c|cccccc} \hline \bar{x} & 0 & 0.2 & 0.4 & 0.6 & 0.8 & 1 \\ \hline P(\bar{x}) & 0.03 & 0.16 & 0.31 & 0.31 & 0.16 & 0.03 \\ \hline \end{array} \]

n = 10:

\[ \begin{array}{c|ccccccccccc} \hline \bar{x} & 0 & 0.1 & 0.2 & 0.3 & 0.4 & 0.5 & 0.6 & 0.7 & 0.8 & 0.9 & 1 \\ \hline P(\bar{x}) & 0.00 & 0.01 & 0.04 & 0.12 & 0.21 & 0.25 & 0.21 & 0.12 & 0.04 & 0.01 & 0.00 \\ \hline \end{array} \]

n = 20:

\[ \begin{array}{c|ccccccccccc} \hline \bar{x} & 0 & 0.05 & 0.10 & 0.15 & 0.20 & 0.25 & 0.30 & 0.35 & 0.40 & 0.45 & 0.50 \\ \hline P(\bar{x}) & 0.00 & 0.00 & 0.00 & 0.00 & 0.01 & 0.04 & 0.07 & 0.12 & 0.16 & 0.18 \\ \hline \end{array} \] \[ \begin{array}{c|ccccccccccc} \hline \bar{x} & 0.55 & 0.60 & 0.65 & 0.70 & 0.75 & 0.80 & 0.85 & 0.90 & 0.95 & 1 \\ \hline P(\bar{x}) & 0.16 & 0.12 & 0.07 & 0.04 & 0.01 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 \\ \hline \end{array} \]

Histograms illustrating these distributions are “Distributions of the Sample Mean”.

Population and population distribution refer to the number and spread of people within a certain area. These two concepts are important for understanding patterns of population growth, urbanization, and regional development.

The population distribution for a discrete variable is:

\[ P(X = x_i) = \frac{\text{number of elements equal to } x_i}{N} \] The population distribution for a continuous variable is:

\[ f(x) \ge 0, \qquad \int_{-\infty}^{\infty} f(x)\, dx = 1 \]

Example for a population {0, 1}:

\[ P(X = 0) = 0.5, \quad P(X = 1) = 0.5 \]

1. Sample Size: The CLT holds true when the sample size is large enough, typically n ≥ 30. The larger the sample size, the closer the sampling distribution of the sample mean will be to a normal distribution.

2. Independence: The samples must be independent of each other.

3. Finite Variance: The population from which samples are drawn must have a finite variance.

The CLT can be mathematically expressed as:

\[ \overline{X} \sim N\left(\mu,\; \frac{\sigma}{\sqrt{n}}\right) \]

where:

• \(\overline{X}\) is the sample mean

• \(\mu\) is the population mean

• \(\sigma\) is the population standard deviation

• \(n\) is the sample size

Example 1: Turtle Shell Widths

Suppose the width of a turtle’s shell follows a uniform distribution with a minimum width of 2 inches and a maximum width of 6 inches. If we take random samples of 30 turtles and calculate the mean shell width each time, the distribution of these sample means will approximate a normal distribution

Example 2: Number of Pets per Family

Consider the number of pets per family in a city follows a chi-square distribution with three degrees of freedom. If we take random samples of 30 families and calculate the mean number of pets each time, the distribution of these sample means will also approximate a normal distribution.

In statistics, when conducting a survey, usually not all population data is known, so a study is typically conducted on a representative sample, and then the conclusions drawn are extrapolated to the entire population. Therefore, the sample proportion is used to estimate the proportion of the entire population. Below we will see how this is done.

The sample proportion is equal to the number of successful cases in a sample divided by the total number of samples. Therefore, the formula to calculate the sample proportion is:

\[ \widehat{p} = \frac{e}{n} \]

where:

• \(\widehat{p}\) : proporsi sampel
  

• \(e\) : jumlah kasus yang berhasil dalam sampel

• \(n\) : jumlah total item data dalam sampel
  

• A company manufactures toys and buys some of its parts from another external company. However, in the batch it purchased, defective parts appeared, so it decided to conduct a statistical study to determine the proportion of parts in good condition and the proportion of defective parts. So, you order a sample of 1,000 units and find 138 defective parts. What is the proportion of parts in good condition in the sample? And what is the proportion of defective parts in the sample?

The number of non-defective parts in the sample is 1000 minus the number of defective parts: \[ e = 1000 - 138 = 862 \]

Answer: \[ \widehat{p} = \frac{e}{n} = \frac{862}{1000} = 0.862 \]

Therefore, the proportion of spare parts in good condition is 86.2%.

Conversely, the proportion of defective spare parts is equal to one minus the proportion of good spare parts:

\[ \widehat{q} = 1 - \widehat{p} = 1 - 0.862 = 0.138 \]

Therefore, the proportion of defective parts in the sample is 13.8%.

The review of probability theory covers essential concepts such as the sample space, event space, probability measure, and axioms of probability. It also includes the calculus of probability, random variables, probability distributions, expectation, moments, and extreme-value distributions. The review is suitable for those who have completed a course on probability and statistics and provides a foundational understanding for further study in related fields.

• Suppose we have a jar that contains 200 green marbles, and 300 blue marbles. If a marble is drawn three times with replacement, what is the probability of drawing at least two green marbles?

Probability of Green

\[ P(\text{Green}) = \frac{\text{number of successful outcomes}}{\text{total number of outcomes}} \]

\[ P(\text{Green}) = \frac{200}{500} = 0.4 \]

Probability of Blue

\[ P(\text{Blue}) = \frac{\text{number of unsuccessful outcomes}}{\text{total number of outcomes}} \]

\[ P(\text{Blue}) = \frac{300}{500} = 0.6 \]

• Sampling Distribution of the Sample Mean

Given:

• Population mean: \(\mu = 13,525\)

• Population standard deviation: \(\sigma = 4,180\)

• Sample size: \(n = 100\)

Mean of the Sampling Distribution

\[ \mu_{\bar{X}} = \mu = 13,525 \]

Standard Error of the Mean

\[ \sigma_{\bar{X}} = \frac{\sigma}{\sqrt{n}} \]

Substitute values:

\[ \sigma_{\bar{X}} = \frac{4,180}{\sqrt{100}} \]

\[ \sigma_{\bar{X}} = 418 \]

Interpretation

This means the sample means will cluster around:

\[ 13,525 \quad \text{with variability (standard error) of} \quad 418. \]

1. Definition: The sampling distribution is the distribution of a statistic (e.g., sample mean) calculated from all possible samples of a fixed size from a population. For example, if we repeatedly sample 50 dolphins from a population and calculate their mean weights, the distribution of these means forms the sampling distribution.

2. Properties: The mean of the sampling distribution of the sample mean equals the population mean (μx̄ = μ). The standard deviation of the sampling distribution (known as the standard error) is given by σx̄ = σ / √n, where σ is the population standard deviation and n is the sample size.

3. Central Limit Theorem (CLT): Regardless of the population’s shape, the sampling distribution of the sample mean approaches a normal distribution as the sample size increases (n > 30). This is crucial for making statistical inferences.

4. Types of Sampling Distributions: Sampling Distribution of the Mean: Focuses on the means of samples. For example, the average weight of dolphins in repeated samples. Sampling Distribution of the Proportion: Focuses on proportions, such as the percentage of dolphins that are black in repeated samples. T-Distribution: Used when the population standard deviation is unknown or the sample size is small.

Sampling distributions are essential for hypothesis testing, constructing confidence intervals, and making predictions about population parameters. For instance, they help determine the likelihood of observing a sample mean under specific conditions, enabling statisticians to draw conclusions about the population.