Conditional Probability and PDF

"The conditional probability of an event B is the probability that the event will occur given the knowledge that an event A has already occurred.

This probability is written P(B|A), notation for the probability of B given A. "

"In the case where events A and B are independent (where event A has no effect on the probability of event B), the conditional probability of event B given event A is simply the probability of event B, that is P(B).

If events A and B are not independent, then the probability of the intersection of A and B (the probability that both events occur) is defined by

P(A and B) = P(A)P(B|A)." Multiplication rule

Ref: http://www.stat.yale.edu/Courses/1997-98/101/condprob.htm

Truncated Distribution
"In statistics, a truncated distribution is a conditional distribution that results from restricting the domain of some other probability distribution. Truncated distributions arise in practical statistics in cases where the ability to record, or even to know about, occurrences is limited to values which lie above or below a given threshold or within a specified range.

For example, if the dates of birth of children in a school are examined, these would typically be subject to truncation relative to those of all children in the area given that the school accepts only children in a given age range on a specific date. There would be no information about how many children in the locality had dates of birth before or after the school's cutoff dates if only a direct approach to the school were used to obtain information."

"

Probability density function for the truncated normal distribution for different sets of parameters. In all cases, a = −10 and b = 10. For the black: μ = −8, σ = 2; blue: μ = 0, σ = 2; red: μ = 9, σ = 10; orange: μ = 0, σ = 10.
Support
PDF
CDF
Mean
Median

"

https://en.wikipedia.org/wiki/Truncated_distribution

Law of Total Probability

Bayes’s theorem

In probability theory and statistics, Bayes’s theorem describes the probability of an event, based on prior knowledge of conditions that might be related to the event. Wikipedia

Formula

	=	events
	=	probability of A given B is true
	=	probability of B given A is true
	=	the independent probabilities of A and B

Ref: https://en.wikipedia.org/wiki/Bayes'_theorem

Bayes Formula for Random Variables:

http://pwp.gatech.edu/ece-jrom/wp-content/uploads/sites/436/2017/08/16_BayesRVs-su14.pdf

Using the above equation for the bayes rule for discrete random variable

Bayes formula for Continuous Random Variable

Using:

Conditional Expectation : Discrete Case

Conditional Expectation : Continuous Case

Ref: https://www.math.arizona.edu/~tgk/464_07/cond_exp.pdf

Gaussian Random Variables:

The PDF:

Ref: https://www.sciencedirect.com/topics/engineering/gaussian-random-variable

Gaussian Random Vector:

Ref: http://statweb.stanford.edu/~kjross/Lec11_1015.pdf

The text and images are from the Internet. References are provided.