chapter3 discrete variable

Example 2 Roll two dice, let X denoted the sum of two dice show number. Solution:
Example 3 Consider the experiment of tossing a coin five times and on each toss observing whether the coin lands with a head or tail on its upward face.
Example 2 : the frequency function is X p -1 0.16 0 1 2 2a/10 3 0.3
a/10 a2
Determine the parameter a? Solution : due to property of the frequency function
Example 1 There are 5 balls in the bag, in which 2 white balls and 3 black balls. Select three balls one time at random manner. Solution: Suppose X={number of selected white balls}
we can define X = 1 when a tail is observed and X = 0 when a head is observed. The definition is arbitrary but it must be fixed before the experiment is started.
3.2 Probability Distributions For Discrete Random Variables
When probabilities are assigned to various outcomes in S , these in turn determine probabilities associated with the values of any particular rv X. the probability distribution of X says how the total probability of 1 is distributed among (allocated to ) the various possible X values.
3 Discistributions
3.1 Random Variables
Definition
For a given sample space S of some experiment, a random variable is any rule that associates a number with each outcome in S.
The Cumulative Distribution Function
Definition The cumulative distribution function (cdf ) F(x) of a discrete rv X with pmf P(x) is defined for every number X by F(x)=P(X≤ x)= y∑xP( y ) y: y ≤ : For any number x, F(x) is the probability that the observed value of X will be at most x.
P(a ≤ X ≤ b)=F(b)-F(a-)
Another View Of Probability Mass Function
Solution: a.
X p 1 1/n 2 1/n 3 1/n …….. …….. n 1/n
b.
1 P ( X = k ) = 1 n
k 1
1 n
k = 1,2,3,
Example 5 The defective rate of automatic product line is p. When a defective products was produced, the automatic product line must be adjusted. Let X denoted the number of good products during two times adjustment. Please determine the probability distribution of X. Solution:
F( x) = P( X ≤ x),
(∞ < x < +∞)
For X a discrete rv, the graph of F(x) will have a jump at every possible value of X and will be flat between possible values. Such a graph is called a step function
A Parameter Of A Probability Distribution Definition Suppose P(x) depends on a quantity that can
be assigned any one of a number of possible values, with each different value determining a different probability distribution. such a quantity is called a a parameter of the distribution . the collection of all probability distributions for different values of the a parameter is called a family of a probability distributions.
0.16 + a/10+ a2+2a/10+ 0.3=1 (a+0.9)(a-0.6)=0 a=0.6
Example 3 Roll two dice, let X denoted the sum of two dice show number. Determine the probability distribution of X ? Solution:
Definition Any random variable whose only possible values are 0 and 1 is called a Bernoulli random variable .
We will often want to define and study several different random variables from the same sample space
Definition The probability distribution or probability
mass function (Pmf) of a discrete rv is defined for every number x by P(x)=P(X=x)
The conditions 1 2 P(x)
P ( X = 0) =
C C
3 3 3 5
1 = 10
P( X
CC = 6 P ( X = 1) = 10 C CC = 3 = 2) = 10 C
3 2 3 5
1 2 2 3 3 5
2
1
The probability distribution function
X p
0 1/10
1 6/10
2 3/10
Example 1 Example 2
Property 1 the cdf is non-decreasing Property 3 satisfies :
Property 2
limF(x) = 0
x→∞
limF(x) = 1
x→+∞
Definition For any two numbers a and b with a ≤ b,
P ( X = k ) = (1 p ) p
k
k = 0,1,2,3,
Example 6 Consider the experiment of tossing a coin five times and on each toss observing whether the coin lands with a head or tail on its upward face. Discuss this experiment with four properties satisfied.
Two Type Of Random Variables Definition A discrete random variable is an rv whose possible values either constitute a finite set or else can be listed in an infinite sequence in which there is a first element, a second element, and so on. A random variable is continuous if its set of possible values consists of an entire interval on the number line.
all possible x
∑ P( x) =1
≥
0