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What is the relationship between the Z-score for Normal Distribution and Continuous Probability?
There is no real relationship. Probabilities for the Normal distribution are extremely difficult to work out. The z-score is a method used to convert any Normal distribution into the Standard Normal distribution so that its probabilities can be looked up in tables easily. There are infinitely many types of continuous probability distributions and the Normal is just one of them.
How does the graph of Fourier Series differ to the graph of Fourier Transform?
You can graph both with Energy on the y-axis and frequency on the x. Such a frequency domain graph of a fourier series will be discrete with a finite number of values corresponding to the coefficients a0, a1, a2, ...., b1, b2,... Also, the fourier series will have a limited domain corresponding to the longest period of your original function. A fourier transforms turns a sum into an integral and as such is a continuous function (with uncountably many values) over the entire domain (-inf,inf). Because the frequency domain is unrestricted, fourier transforms can be used to model nonperiodic functions as well while fourier series only work on periodic ones. Series: discrete, limited domain Transform: continuous, infinite domain.
What is the formula for figuring out probability?
For a discrete distribution, if n trials are carried out and if x of them are favourable to the event that you are interested in, then the probability is x/n. Note that it may be possible to work out n and x on theoretical grounds rather than actually doing the experiment. The formula gets more complicated for continuous variables. For continuous variables, it is the area under the probability density curve over the domain of interest.
What did Kepler notice about the speed of planets as they orbit the sun?
He noted that the speed of the planets changed, moving faster as they approached the Sun, and slowing down as they moved away. His work paved the way for Newton to develop calculus to describe the overall effects of minuscule but continuous changes.
Why imaginary numbers are used?
Originally, they were invented to provide solutions to algebraic equations, which would otherwise have no solution. Through the work of Euler, Gauss and others, the usefulness of imaginary and complex numbers in applications of periodic motion and waves was recognized. See related links.
