What is the difference between probability distribution and sampling distribution?

Step 1. Probability Distribution ($P(X)$):
A probability distribution describes the likelihood of each possible outcome of a random variable. It provides information about the probabilities associated with different values of the variable. The probabilities are assigned to specific values or ranges of values, and they sum up to 1.
For example, let's consider a fair six-sided die. The probability distribution for this die would be denoted as $P(X)$, where $X$ represents the random variable denoting the outcome of rolling the die. The probability distribution for a fair die is given by:
$P(X=x)=\frac{1}{6}\text{for }x\in \{1,2,3,4,5,6\}$
This probability distribution states that each outcome has an equal probability of $\frac{1}{6}$.
Step 2. Sampling Distribution ($\hat{P}(X)$):
A sampling distribution is associated with a statistic, such as the sample mean or sample proportion, calculated from multiple random samples of the same size taken from a population. It describes the distribution of the statistic's values across all possible samples of the same size.
For example, let's consider the heights of students in a school population. We want to estimate the mean height of all students using a sample. If we take multiple random samples of the same size, calculate the mean height for each sample, and plot the distribution of these sample means, it would represent the sampling distribution of the sample mean.
The sampling distribution is denoted as $\hat{P}(X)$, where $X$ represents the random variable associated with the statistic of interest. In this case, $\hat{P}(X)$ would represent the distribution of sample means.
It is important to note that the sampling distribution is centered around the population parameter being estimated and its spread is influenced by both the sample size and the population distribution.
Finally, a probability distribution describes the likelihood of outcomes for a single random variable, while a sampling distribution describes the distribution of values for a statistic across multiple samples taken from a population.

The $probability$ $distribution$ refers to the distribution of probabilities for all possible outcomes of a random variable. It describes the likelihood of each outcome occurring.
On the other hand, the $sampling$ $distribution$ is the distribution of a statistic calculated from multiple samples of the same size taken from a population. It describes the behavior of the statistic as sample size increases.
In summary, the $probability$ $distribution$ deals with the probabilities of individual outcomes, while the $sampling$ $distribution$ deals with the distribution of a statistic calculated from multiple samples.

Result:
The $probability$ $distribution$ focuses on the likelihood of individual outcomes, while the $sampling$ $distribution$ focuses on the distribution of sample statistics obtained from repeated sampling.
Solution:
A $probability$ $distribution$ refers to the distribution of probabilities associated with the possible outcomes of a random variable. It describes the likelihood of each possible outcome occurring. The probability distribution is often represented by a probability mass function (PMF) for discrete random variables or a probability density function (PDF) for continuous random variables.
On the other hand, a $sampling$ $distribution$ refers to the distribution of a sample statistic (such as the mean or standard deviation) that is calculated from multiple samples of the same size taken from a population. In other words, it represents the distribution of sample statistics that would be obtained if we repeatedly drew samples from the same population.
To illustrate the difference, let's consider an example. Suppose we have a population of students and we are interested in their heights. The probability distribution would describe the likelihood of each possible height value for an individual student in the population. It might tell us that the probability of a student being exactly 170 cm tall is 0.05, while the probability of being 180 cm tall is 0.10.
Now, if we were to take multiple random samples of, let's say, 30 students each from the population and calculate the mean height for each sample, we would obtain a sampling distribution of the sample means. This distribution would represent the distribution of all possible sample means that could be obtained by taking samples of size 30 from the population. It provides information about the variability and central tendency of the sample means.