Download 6.2. Probability Distribution (I): Discrete Random Variable:

6.2. Probability Distribution
Definition (probability distribution): a function describes how
probabilities are distributed over the values of the random
variable.
(I): Discrete Random Variable:
Example A (continue):
Suppose the probability for the outcomes in the gamble example is
1
= P ( X = 3) = P (Y = 30 )
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2
P ({ Lose }) = = P ( X = −4) = P (Y = −40 )
3
1
P ({Tie}) = = P ( X = 0) = P (Y = 0)
6
P ({Win}) =
Let f x (x) be some function corresponding to the probability of the gambling
outcomes for random variable X, defined as
f x (3) = P ( X = 3) =
1
6
f x (−4) = P ( X = −4) =
f x (0 ) = P ( X = 0 ) =
2
3
1
6
f x (x) is referred as the probability distribution of random variable X.
Similarly, the probability distribution f y (x) of random variable Y is
f y ( 30 ) = P ( Y = 30 ) =
1
6
f y ( − 40 ) = P ( Y = − 40 ) =
f y ( 0 ) = P (Y = 0 ) =
1
1
6
2
3
Required conditions for a discrete probability distribution:
Let a1 , a 2 ,K , a n ,K be all the possible values of the discrete random
variable X. Then, the required conditions for f (x) to be the discrete
probability distribution for X are
(a)
f ( ai ) ≥ 0, for every i.
(b)
∑ f (a ) = f (a ) + f (a ) +L+ f (a ) +L= 1
i
1
2
n
i
Example A (continue):
In the gambling example, f x (x) is a discrete probability distribution for the random
variable X since
(a)
f x (3) ≥ 0, f x (−4) ≥ 0, and f x (0) ≥ 0 .
(b)
f x (3) + f x (−4) + f x (0) = 1 .
Similarly, f y (x) is also a discrete probability distribution for the random variable Y.
Note: the discrete probability distribution describes the probability
of a discrete random variable at different values.
(II): Continuous Random Variable:
For a continuous random variable, it is impossible to assign a probability to every
numerical value since there are uncountable number of values in an interval. Instead,
the probability can be assigned to a small interval. The probability density function
can describe how the probability distributes in the small interval.
Example B (continue):
In the delay flight time example, suppose the probability of being late within 0.5
hours is two times of the one of being late more than 0.5 hour, i.e.,
P(0 ≤ Z ≤ 0.5) =
2
1
and P(0.5 < Z ≤ 1) = .
3
3
Then, the probability density function f1 ( x) for the random variable Z is
2
0.0
0.5
f1(x)
1.0
1.5
4
2
f1 (x) = , 0 ≤ x ≤ 0.5; f1 (x) = , 0.5 < x ≤ 1.
3
3
0.0
0.2
0.4
0.6
0.8
1.0
x
The area corresponding to the interval is the probability of the random variable Z
taking values in this interval. For example, the probability of the flight time being late
within 0.5 hour (the random variable Z taking value in the interval [0,0.5]). is
0.5
4
2
P(Theflight time being late within0.5 hour)= P(0 ≤ Z ≤ 0.5) = ∫ f1 (x)dx = * 0.5 = .
3
3
0
Similarly, the probability of the flight time being late more than 0.5 hour (the random
variable Z taking value in the interval (0.5,1]). is
1
2
1
P(Theflight time being late morethan0.5 hour)= P(0.5 < Z ≤ 1) = ∫ f1 (x)dx = * 0.5 = .
3
3
0.5
On the other hand, If the probability of being late within 0.5 hours is the same as the
one of being late more than 0.5 hour, i.e.,
P(0 ≤ Z ≤ 0.5) = P(0.5 < Z ≤1) =
1
,
2
then, the probability density function f 2 ( x) for the random variable Z is
f 2 ( x ) = 1, 0 ≤ x ≤ 1 .
Note that the probability density function corresponds to the probability of the random
variable taking values in some interval. However, the probability density
function evaluated at some value, not like the probability distribution,
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can not be used to describe the probability of the random variable Z
taking this value.
Required conditions for a continuous probability density:
Let the continuous random variable Z taking values in [a,b]. Then,
the required conditions for f (x) to be the continuous probability
density for Z are
(a)
f ( x ) ≥ 0, a ≤ x ≤ b .
b
(b)
∫
f ( x )dx = 1
a
d
Note: P (c ≤ Z ≤ d ) =
∫ f ( x)dx, a ≤ c ≤ d ≤ b . That
is, the
c
area under the graph of f (x) corresponding to a given interval is the
probability of the random variable Z taking value in this interval.
Example B (continue):
In the flight time example, f1 ( x) is a continuous probability density for the random
variable Z since
(a)
f1(x) ≥ 0, 0 ≤ x ≤ 1 .
0 .5
(b)
∫
0
4
dx +
3
1
2
∫0 .5 3 dx = 1 .
Similarly, f 2 ( x) is also a continuous probability density for the random variable Z.
Online Exercise:
Exercise 6.2.1
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