Introduction:
Studying the properties and applications of probability distributions is essential for understanding and analyzing random phenomena in various fields such as statistics, economics, physics, and engineering. Probability distributions provide a mathematical framework to describe the likelihood or probability of different outcomes.

Types of Probability Distributions:
There are several types of probability distributions, each with its own properties and applications. Some commonly studied distributions include:

1. Discrete Probability Distributions:
- Discrete distributions model random variables with a finite or countable set of possible outcomes.
- Examples include the Bernoulli, Binomial, and Poisson distributions.
- Bernoulli distribution models a single trial with two possible outcomes (success or failure).
- Binomial distribution models the number of successes in a fixed number of independent trials.
- Poisson distribution models the number of events occurring in a fixed interval of time or space.

2. Continuous Probability Distributions:
- Continuous distributions model random variables with an infinite set of possible outcomes within a range.
- Examples include the Normal, Exponential, and Uniform distributions.
- Normal distribution is widely used due to the Central Limit Theorem and its symmetrical bell-shaped curve.
- Exponential distribution models the time between events in a Poisson process.
- Uniform distribution models the equally likely outcomes within a specified range.

Properties of Probability Distributions:
Understanding the properties of probability distributions is crucial for their effective study. Some important properties include:

1. Probability Density Function (PDF) or Probability Mass Function (PMF):
- PDF for continuous distributions and PMF for discrete distributions describe the probability of each possible outcome.
- PDF/PMF integrates/sums to 1 over the range of possible outcomes.

2. Cumulative Distribution Function (CDF):
- CDF gives the probability that a random variable takes on a value less than or equal to a given value.
- CDF is obtained by integrating the PDF/PMF.

3. Expected Value and Variance:
- Expected value represents the average value of a random variable and is denoted by E(X).
- Variance measures the spread or dispersion of a distribution and is denoted by Var(X).

4. Moments and Moment Generating Function:
- Moments provide a way to summarize the shape and properties of a distribution.
- Moment generating function (MGF) is a mathematical technique to derive moments of a distribution.

Applications of Probability Distributions:
Probability distributions have numerous applications in various fields. Some common applications include:

1. Statistical Analysis:
- Probability distributions serve as the basis for statistical inference, hypothesis testing, and estimation.
- They help analyze and interpret data, make predictions, and evaluate the significance of results.

2. Risk Assessment and Decision-Making:
- Probability distributions are used to model and analyze risks in finance, insurance, and project management.
- Decision-making under uncertainty is facilitated by understanding the probabilities of different outcomes.

3. Reliability Engineering:
- Probability distributions are employed to model the failure rates and lifetimes of components and systems.
- They help estimate reliability, availability, and maintainability of engineering