HOOKE'S LAW Introduction In physics, Hooke's Law is one of the fundamental principles governing how objects deform under external forces . Named after the 17th-century British physicist Robert Hooke, this law provides a crucial understanding of the behavior of elastic materials, such as springs and rubber bands. Whether stretching a rubber band or compressing a spring, Hooke's Law helps explain what happens when forces act on these materials. What is Hooke's Law: Hooke's Law states that the force F needed to extend or compress a spring by some distance x is proportional to that distance. Mathematically, it is expressed as: F= -kx Here k represents the spring constant, which is the measure of the stiffness of the spring, and x is the displacement from the displacement position. The negative sign indicates that the force exerted by the spring is in the opposite direction of the displacement. Understanding the Spring Constant: The spring constant k is a critical co...
PROBABILITY AND RANDOM SIGNALS
INTRODUCTION TO PROBABLITY:
Probability is a number that describes a set.
▪ The higher the number, the more probability there is.
▪ In this sense probability is like a quantity that measures a physical
phenomenon
▪ Methods used for assigning probabilities
▪ Personal Approach
▪ Relative Frequency Approach
▪ Classical Approach
▪ Study of probability refers to an experiment consisting of a procedure and
observations.
▪ In random variables, each observation corresponds to one or more
numbers.
▪ In stochastic processes, each observation corresponds to a function of time.
▪ The word stochastic means random.
▪ In stochastic /random processes, we study random functions of time.
▪ Almost all practical applications of probability involve multiple observations
taken over a period of time.
SET THEORY:
Set theory is the mathematical theory of well-determined collections,
called sets, of objects that are called members, or elements, of the set.
• Set Theory is a branch of mathematics that investigates sets and their
properties.
• The mathematical basis of probability is the theory of sets
- Basic sets
▪ Operations
▪ Venn Diagram
▪ Properties
▪ Conditional probability
▪ Independent events
▪ Baye's formula
▪ Discrete and continuous random variables
▪ Distributions
▪ Density functions
PROBABILITY DISTRIBUTIONS
▪ Binomial Distribution
▪ Poisson Distribution
▪ Hyper geometric Distribution
▪ Normal Distribution
▪ Uniform Distribution
▪ Exponential Distribution
STATISTICS:
• Statistics is the science concerned with developing and studying methods
for collecting, analyzing, interpreting and presenting empirical data.
• Statistics studies methodologies to gather, review, analyze and draw
conclusions from data.
▪ Mean
▪ Variance
▪ Standard deviations
▪ Moments and
▪ Moment generating functions
REGRESSION AND CURVE FITTING:
Regression takes a group of random variables and tries to find a
mathematical relationship between them.
Curve fitting is the process of specifying the model that provides the best
fit to the curve in your data.
▪ Linear regression
▪ Curve fitting
▪ Limits theorems
STOCHASTIC PROCESSES:
• A stochastic process is a system which evolves in time while
undergoing chance fluctuations.
▪ First and second order characteristics
▪ Applications
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