Final answer:
Brownian motion has distinct properties such as starting at zero with independent increments and a normal distribution at each time. The joint distribution represents the probabilities of multiple variables, and when it's bivariate normal, it speaks to both variables being normally distributed with each linear combination also normally distributed.
Step-by-step explanation:
Brownian motion, also known as a Wiener process, is a path taken by a particle that is under the influence of random forces. Its properties include (1) it starts at zero, (2) has stationary and independent increments, (3) has a normal distribution with a mean of zero and a variance of t at time t, and (4) it has continuous paths.
The joint distribution refers to the probability distribution that encapsulates the likelihood of two or more random variables taking on various combinations of values. Understanding joint distribution is crucial when dealing with pairs or sets of random variables and allows for the assessment of their collective behavior.
For bivariate normal distributions, the characteristics include (1) each variable is normally distributed, (2) any linear combination of the variables is also normally distributed, and (3) the distribution is defined by the mean and covariance matrix of the variables. The implications of a joint distribution being normal are significant for statistical analysis and modeling, allowing for the use of a wide array of statistical tools and simple calculation of probabilities and expectations.