STA220H5 Lecture Notes - Lecture 3: Probability Density Function, Probability Distribution, Conditional Probability
(3.5-3.6)
Conditional probability formula
To find the conditional probability that event A occurs given that event B occurs,
divide the probability that both A and B occur by the probability that B occurs; that
is,
P (A/B) = P (AnB)/P(B)
[We assume that P(B) is not equal to 0.]
Multiplicative rule of probability
P (AnB) = P (A) P (B/A) or, equivalently, P(AnB) = P(B)P(A/B)
Events A and B are independent events if the occurrence of B does not alter the
probability that A has occurred; that is, events A and B are independent if
P (A/B) = P (A)
When events A and B are independent, it is also true that
P (B/A) = P(B)
Events that are not independent are said to be dependent.
Probability of intersection of two independent events
If events A and B are independent, then the probability of the intersection of A and B
equals the product of the probabilities of A and B; that is,
P(AnB)= P(A)P(B)
The converse is also true: if P(AnB) =P(A)P(B), then events A and B are
independent.
5.1-5.2
The probability distribution for a continuous random variable, x, can be represented
by a smooth curve – a function of x, denoted f ( x ) .
The curve is called a density function or frequency function.
The probability that x falls between two values, a and b, i.e., P ( a < x < b ) , is the
area under the curve between a and b.
Probability distribution for a uniform random variable x
Probability density function: f ( x ) = 1 / d – c ( c < x < d )
Mean: u = c + d / 2 standard deviation: a = d – c / 12
P ( a < x < b ) = ( b - a ) / ( d - c ) , c < a < b < d
(6.1-6.2)
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Document Summary
To find the conditional probability that event a occurs given that event b occurs, divide the probability that both a and b occur by the probability that b occurs; that is, [we assume that p(b) is not equal to 0. ] P (anb) = p (a) p (b/a) or, equivalently, p(anb) = p(b)p(a/b) Events a and b are independent events if the occurrence of b does not alter the probability that a has occurred; that is, events a and b are independent if. When events a and b are independent, it is also true that. Events that are not independent are said to be dependent. If events a and b are independent, then the probability of the intersection of a and b equals the product of the probabilities of a and b; that is, The converse is also true: if p(anb) =p(a)p(b), then events a and b are independent.
