The normal distribution is widely used in statistics and is characterized by its bell-shaped curve. It is symmetric, with the mean, median, and mode all being equal. The sampling distribution of the mean, when the sample size is large, follows a normal distribution. This is known as the Central Limit Theorem.


The t-distribution is similar to the normal distribution, but with heavier tails. It is used when the population standard deviation is unknown and the sample size is small. The shape of the t-distribution depends on the degrees of freedom, which is determined by the sample size.


The chi-square distribution is used when working with categorical data and testing for independence or goodness-of-fit. It is characterized by its skewed shape and is positively skewed. The shape of the chi-square distribution depends on the degrees of freedom.


The F-distribution is used for hypothesis testing in analysis of variance (ANOVA) and regression analysis. It is right-skewed and characterized by its two parameters, degrees of freedom for the numerator and denominator. The F-distribution is used to compare the variances of two or more populations.


The exponential distribution is used for modeling the time between events in a Poisson process. It is characterized by its decreasing exponential shape and is skewed to the right. The exponential distribution is often used in reliability analysis and queuing theory.


The binomial distribution is used when dealing with two possible outcomes, such as success or failure. It is characterized by its discrete nature and is used to model the number of successes in a fixed number of independent Bernoulli trials.


The Poisson distribution is used to model the number of events occurring in a fixed interval of time or space. It is characterized by its discrete nature and is often used in queuing theory, reliability analysis, and epidemiology.

Each of these sampling distributions has its own characteristics and applications. Understanding these distributions is crucial in statistical analysis and hypothesis testing.