### Distribution (Continuous)

continuous uniform distribution: A continuous random variable whose probability distribution is the uniform distribution is often called a uniform random variable. If we know nothing about a random variable apart from the fact that it has a lower and an upper bound, then a uniform distribution is a natural model

• mean: $\frac{u+l}{2}$
• variance: $\frac{(u-l)^2}{12}$

exponential distribution: We assume that failures form a Poisson process in time; then the time to the next failure is exponentially distributed.

• mean: $\frac{1}{\lambda}$
• variance: $\frac{1}{\lambda ^ 2}$

### Normal Distribution

a probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean

$$p(x)=(\frac{1}{\sqrt{2\pi}\sigma})\text{exp}(\frac{-(x-\mu)^2}{2\sigma^2})$$

• mean: $\mu$
• variance: $\sigma ^2$
• 68% data within $\sigma$, 95% data within $2\sigma$, 99% data within $3\sigma$
• A continuous random variable is a normal random variable if its probability density function is a normal distribution.
• another name for normal distribution is Gaussian distributions.
• central limit theorem (CLT): under some not very worrying technical conditions, the sum of a large number of independent random variables will be very close to normal.

standard normal distribution:

$$p(x)=(\frac{1}{\sqrt{2\pi}})\text{exp}(\frac{-x^2}{2})$$

• mean: $0$
• variance: $1$
• A continuous random variable is a standard normal random variable if its probability density function is a standard normal distribution.
• Any probability density function that is a standard normal distribution in standard coordinates is a normal distribution.

binomial distribution approximation:

when $N$ is huge, we can approximate binomial distribution with normal distribution to reduce calculation cost.

Assume h follows the binomial distribution with parameters p and q. Write:

$$x=\frac{h-Np}{\sqrt{Npq}}$$

The, for large N, the probability distribution $P(x)$ can be approximated by the probability density function:

$$P(\{x\in [a,b]\})\approx \int^b_a(\frac{1}{\sqrt{2\pi}})\text{exp}(\frac{-x^2}{2})$$

### Experiment

population and sample: if we could have seen everything, is the population. I will write populations like random variables with capital letters to emphasize we don’t actually know the whole population. The data we actually have is the sample.

sample mean: the mean from sample, usually notated as: $X^{(N)}$: sample mean when sample size is $N$

• it is a random variable
• $\mathbb{E}[X^{(N)}]=\text{popmean(\{X\})}$
• $var[X^{(N)}]=\frac{\text{popsd(\{X\})}^2}{N}$
• $std[X^{(N)}]=\frac{\text{popsd(\{X\})}}{\sqrt{N}}$

confident interval:

• confidence interval for a population mean: Choose some fraction f, An f confidence interval for
a population mean is an interval constructed using the sample mean. It has the property that for that fraction f of all samples, the population mean will lie inside the interval constructed from each sample’s mean.
• centered confidence interval for a population mean: Choose some$0<\alpha<0.5$. A $1-2\alpha$ centered confidence interval for a population mean is an interval $[a,b]$; b constructed using the sample mean.

unbiased standard deviation: use to estimate the population standard deviation

$$\text{stdunbiased(\{x\})}=\sqrt{\frac{\sum_i(x_i-\text{mean}(\{x\})^2)}{N-1}}$$

standard error: The standard deviation of the estimate of the mean

$$\text{stderr}(\{x\})=\frac{\text{stdunbiased(\{x\})}}{\sqrt{N}}$$

### Random variable distribution

we learn two distribution for random variable for far: the t-distribution and normal distribution, depended on the sample size, if $n<30$, we use t-distribution, otherwise use normal distribution

t-distribution:

$$T=\frac{\text{mean}(\{x\})-\text{popmean(\{X\})}}{\text{stderr(\{x\})}}$$

degree of freedom: $N-1$

normal distribution:

$$Z=\frac{\text{mean}(\{x\})-\text{popmean(\{X\})}}{\text{stderr(\{x\})}}$$

degree of freedom: $N$