Agenda
Probability Distributions
Random Variables
Probability Distributions
Discrete Probability Distributions
Two common discrete distributions
Common continuous distributions
Probability Density Functions
Probability Mass Functions for Discrete distributions
Probability Density Functions for Continuous Distributions
Probability Mass Functions
Binomial
\( f(x)=\textstyle {n \choose k}\, p^k (1-p)^{n-k} \quad where \)
n= number of trials,
k= number of successes
Poisson
\( f(x)=\frac{\lambda^k e^{-\lambda}}{k!} \quad where \)
\( \lambda \) = is the average number of events per interval (event rate)
k= number of occurrences
Probability Density Functions
Uniform
\[ f(x)=\begin{cases} \frac{1}{b - a} & \text{for } x \in [a,b] \\ 0 & \text{otherwise} \end{cases} \]
Normal
\[ f(x)=\frac{1}{\sigma\sqrt{2\pi}}\, e^{-\frac{(x - \mu)^2}{2 \sigma^2}} \]
Probabilities in Distributions
Distributions in R
Example - Normal Distributions
\( N(150,5) \) is a normal distribution where \( \mu \)= 150 and \( \sigma \)= 5
dnorm() for density function
x=seq(120,180, 1)
d150= dnorm(150, mean = 150, sd=5)
plot(x, props, type = 'l')
points(150, d150)
\( N(150,5) \) is a normal distribution where \( \mu \)= 150 and \( \sigma \)= 5
\( N(150,5) \) is a normal distribution where \( \mu \)= 150 and \( \sigma \)= 5
qnorm() to get the quantiles - inverse of pnorm
quartile.p=seq(0.25, .75, 0.25)
(quantiles=qnorm(quartile.p, mean=150, sd = 5))
[1] 146.6276 150.0000 153.3724
props=dnorm(x, 150, 5)
plot(x, props, type = 'l')
points(quantiles, dnorm(quantiles, 150, 5))
Try to use these functions in binomial distributions
dbinom, pbinom, qbinom, rbinom
y<-1:30
b.dist<-plot(y, dbinom(y, 30, 0.5), type = "h", ylab = 'Binomial Dist', main = '# of heads in 30 coin flips')
Confidence Intervals
Running example
The Central Limit Theorem (reminder)
- The Central Limit Theorem (CLT) is one of the most important theorems in statistics
- The useful way to think about the CLT is that \( \bar X_n \) is approximately \( N(\mu, \sigma^2 / n) \). That is, the sample mean, \( \bar X \), is approximately normal with mean (\( \mu \)) and sd (\( \sigma / \sqrt{n} \))
\[ \mu_{\bar{x}} = \mu, \ \sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}} \]
Central Limit Theorem
According to the central limit theorem, the average height for samples of 237 observations follow a normal distribution of N (\( \mu, \sigma/\sqrt{237} \) ).
Our sample mean (172.4) is one of the outcomes in this distribution, we just don't know where it actually is.
Confidence Intervals
Area under the curve can be calculated in reference to the distance in standard deviation from the mean
Confidence Intervals (Contd.)
- For a conf. level of C (say 95%), the confidence interval should be z many standard errors (\( \sigma/\sqrt{n} \)) far out in each tail.
- z = quantile(1 - (1-C)/2).
- (1-C) is also known as \( \alpha \), so z=qnorm(1 - \( \alpha/2 \))
C.I. with known variance (z-test)
The 95% confidence level would imply the 97.5th percentile of the normal distribution at the upper tail. Therefore, 1.96 (\( z_{\alpha/2} \) ) is given qnorm(.975)
Confidence intervals with unknown variance (or small n)
\( t_{\alpha/2, n-1}=qt(1-\alpha/2, n-1 ) \)
t-test
t-test (shortcut)
proportion test
proportion test (shortcut)
Hypothesis Test
Hypothesis Testing
Decisions based on the p value
One sided vs Two sided
Hypothesis Tests with Known Variance (\(\sigma\))
Hypothesis test for Proportions
One-sided vs Two-sided for proportions
Type I and Type II errors
- There are two competing hypotheses: the null and the alternative.
In a hypothesis test, we make a decision about which might be true, but our choice might be incorrect.
A Type I Error is rejecting the null hypothesis when \( H_0 \) is true.
A Type II Error is failing to reject the null hypothesis when \( H_A \) is true.