2. S t a t i s t i c a l i n f e r e n c e
1
Uncertainties
3. Definition of random variables
Random variables are variables that
can take on different values as
outcomes of a random phenomenon.
They represent numerical outcomes of
a random experiment.
4. Discrete vs. continuous
random variables
Discrete random variables take on a finite
or countably infinite number of distinct
values. Example: the number of heads in
a series of coin flips (discrete).
5. Discrete vs.
continuous random
variables
Continuous random variables
can take on any value within a
specified range.
Example: the height of
individuals in a population
(continuous), or the time until
a light bulb fails (continuous)
6. Statistical vs. Systematic Uncertainties
Statistical uncertainty arises from the inherent randomness in data or
measurements.
Systematic uncertainty arises from biases or errors that consistently affect
measurements in the same way.
Examples:
Statistical uncertainty might arise from sampling variability in survey data,
while systematic uncertainty might arise from measurement instrument
calibration errors.
Distinguishing between statistical and systematic uncertainties is crucial for
accurately assessing the reliability and validity of data and making informed
decisions based on the data.
7. Expected value
The expected value of a
discrete random variable
represents the average
outcome of the variable
and is calculated as the
sum of each outcome
multiplied by its
probability.
Variance
The Variance measures the
spread or variability of the
random variable around its
mean.
8. Uncertainty propagation
When functions of random variables are involved, uncertainties in the inputs
propagate to uncertainties in the outputs. Understanding this propagation is
essential for assessing the overall uncertainty in a model or system.
In linear functions, uncertainties propagate straightforwardly by scaling the
uncertainties in the input variables by the coefficients in the function.
When input random variables are correlated, their covariance must be taken into
account in addition to their individual variances to accurately propagate
uncertainties.
Nonlinear Functions of Random Variables
Challenges in propagating uncertainties for nonlinear functions: Nonlinear functions
of random variables introduce complexities in uncertainty propagation because they
do not scale linearly with input uncertainties.
11. Definition of covariance and correlation
Covariance measures the degree to which two random variables change
together. Correlation is a standardized measure of covariance, representing
the strength and direction of the linear relationship between two random
variables.
Interpretation of covariance and correlation
Positive covariance indicates that when one variable is above its mean, the
other tends to be above its mean as well.
Negative covariance indicates the opposite.
Correlation ranges from -1 to 1, where 1 indicates a perfect positive linear
relationship, -1 indicates a perfect negative linear relationship, and 0
indicates no linear relationship.
