Probability and Statistics, including Distributions and Hypothesis Testing

Probability and statistics play a significant role in many fields, including physics and engineering. These disciplines often require the understanding and analysis of complex data sets and making predictions based on that data. In this article, we will focus on the basic concepts and equations of probability and statistics, including distributions and hypothesis testing, as they are applied in physics and engineering.
DistributionsA distribution is a representation of the frequency or probability of occurrence of a particular outcome or set of outcomes in a random event. In probability and statistics, several types of distributions are commonly used:
Normal DistributionThe normal distribution, also known as the Gaussian distribution, is one of the most widely used distributions in physics and engineering. It is a continuous probability distribution with a bell-shaped curve that is symmetrical about its mean. The normal distribution formula is given by:
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f(x)=12πσ2e−(x−μ)22σ2f(x) = \frac{1}{\sqrt{2\pi \sigma^2}} \text{e}^{-\frac{(x-\mu)^2}{2\sigma^2}}f(x)=2πσ2​1​e−2σ2(x−μ)2​
Where: 
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μ =mean of dist.σ=std. dev\mu\ = mean \: of \: dist. \\ \sigma = std. \: devμ =meanofdist.σ=std.dev
Binomial DistributionThe binomial distribution is used to represent the number of successful outcomes in a fixed number of independent Bernoulli trials. The binomial distribution formula is given by:
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P(x)=(nx)pxqn−x=n!x!(n−x)!pxqn−xP(x) = \binom{n}{x}p^xq^{n-x} = \frac{n!}{x!(n-x)!} p^xq^{n-x}P(x)=(xn​)pxqn−x=x!(n−x)!n!​pxqn−x
Where: 
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n=number of trials (samples)x=number of successes desiredp=probability of success in one trialq=1−pn = number \: of \: trials \: (samples) \\ x = number \: of \: successes \: desired \\ p = probability \: of \: success \: in \: one \: trial \\ q = 1-pn=numberoftrials(samples)x=numberofsuccessesdesiredp=probabilityofsuccessinonetrialq=1−p
Poisson DistributionThe Poisson distribution is used to represent the number of events occurring in a fixed interval of time or space. It is often used to model the frequency of rare events. The Poisson distribution formula is given by:
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f(x)=λxx!e−λf(x) = \frac {\lambda^x} {x!} e^{-\lambda} f(x)=x!λx​e−λ
Where:
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λ=avg. number of eventsx=Poisson random variable\lambda = avg. \: number \: of \: events \\ x = Poisson \: random \: variable  λ=avg.numberofeventsx=Poissonrandomvariable
Exponential DistributionThe exponential distribution is used to model the time between events in a Poisson process. The equation for the exponential distribution is given by:
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f(x)=λe−λxf(x) = \lambda \text{e}^{-\lambda x}f(x)=λe−λx
Where: 
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λ=rate of occurrence of eventsx=Time between events\lambda = rate \: of \: occurrence \: of \: events \\ x = Time \: between \: events λ=rateofoccurrenceofeventsx=Timebetweenevents
Hypothesis TestingHypothesis testing is a statistical method used to test a claim or hypothesis about a population parameter. In physics and engineering, hypothesis testing is often used to test the validity of a model or to determine if a relationship exists between two variables. 
The basic steps in hypothesis testing are:
State the null hypothesis and the alternative hypothesis.
Choose a significance level, usually denoted by alpha
Calculate the test statistic and the p-value.
Compare the p-value to the significance level.
Make a decision about the null hypothesis and state a conclusion.
t-statisticThe most commonly used test statistic in hypothesis testing is the t-statistic, which is used to test the mean of a normally distributed population. The equation for the t-statistic is given by:
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t=xˉ−μsnt = \frac{\bar{x} - \mu}{\frac{s}{\sqrt{n}}}t=n​s​xˉ−μ​
Where: 
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xˉ=sample meanμ=population means=sample std. dev.n=sample size\bar{x}  = sample \: mean \\ \mu = population \: mean \\ s = sample \: std. \: dev. \\ n = sample \: sizexˉ=samplemeanμ=populationmeans=samplestd.dev.n=samplesize
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The p-value is the probability of observing a test statistic as extreme or more extreme than the one calculated, given that the null hypothesis is true.
If the p-value is less than the significance level:
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p<αp < \alphap<α
then the null hypothesis is rejected and the alternative hypothesis is accepted. This means that there is sufficient evidence to support the claim that the population parameter is different from what was stated in the null hypothesis. On the other hand, if the p-value is greater than the significance level then the null hypothesis is not rejected and it is concluded that there is not enough evidence to support the claim.
Applications of Hypothesis TestingHypothesis testing is used in many fields, including physics and engineering, to test the validity of models or to determine relationships between variables. 
For example, in physics, hypothesis testing can be used to determine if a new model for predicting the behavior of a physical system is accurate. 
In engineering, hypothesis testing can be used to determine if there is a relationship between two variables, such as the strength of a material and its composition.
It is important to note that hypothesis testing is only a starting point and additional analysis may be required to confirm the results. It is also important to choose the appropriate test and significance level based on the problem being solved and the data being analyzed.
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