1. 3 Discrete Random Variables and Probability Distributions

2. 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.
Use rv in place of random variable. Random variables are denoted by uppercase letters, such as X and Y , x to represent some particular value of the corresponding random variable.

3. 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.
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

4. 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}
Example 2 Roll two dice, let X denoted the sum of two dice show number.
Solution:

5. 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.

6. 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.

7. 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.
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)

8. X x1 x2 … p p1 p2
The probability distribution/ The frequency function

9. 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, determine the probability distribute function of the number of selected balls being white
Solution: suppose X={number of selected white balls}
The probability distribution function X
p / / /10

10. Example 2 : The frequency function is
p a/10 a a/
Determine the parameter a?
Solution : due to property of the frequency function
a/10+ a2+2a/ =1
(a+0.9)(a-0.6)= a=0.6

11. Example 3 Roll two dice, let X denoted the sum of two dice show number
Example 3 Roll two dice, let X denoted the sum of two dice show number. Determine the probability distribution of X ?
Solution: X 2 3 4 5 6 7 8 9 10 11 12 p
1/36
2/36
3/36
4/36
5/36
6/36

12. Example 4 Someone use n keys to open a door, but only one key can open the door, so he try one by one. Let X denoted try times till the door be opened. Please write the probability distribution of X under follow condition.
a. The keys that could not open the door will put away.
b. The keys that could not open the door will not be put away.

13. Solution: a. X 1 2 3
…….. n p
1/n b.

14. Example 5 The defective rate of automatic product line is p
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:

15. 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.

16. 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.

17. 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
For any number x, F(x) is the probability that the observed value of X will be at most x.

18. Property 1 the cdf is non-decreasing
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
Example 1 P105 11
Example 2
Property 1 the cdf is non-decreasing
Property 2 right continuous
Property 3 satisfies :

19. Definition
Another View Of Probability Mass Function: (omit)

20. 3.3 Expected Values Of Discrete Random Variables
The Expected Values Of X
Definition: Let X be a discrete rv with set of possible values D and pmf p(x). the expected values or mean value of X, is
Provided that ; if the sum diverges, the expectation is undefined.

21. Example: Determine the expected number of spots Y that show when a fair die is rolled in an honest manner.
average
Example:
Expectation of a Geometric Random Variable
X ~ G(p),

23. Example : One people use n keys to open a particular door, and only one key can open the door, but he does not know which one, he tries one by one, till the door opened. Let X denote the number of his tries. Please calculate EX on the condition of
(1) the keys unopened the door are put away;
(2) the keys unopened the door are not put away.
Solution: (1) the probability function is
P(X=k)=1/n, k=1,2,…,n
So,

24. (2) the probability function is

25. Example : In a shooting game, each shooter may shoot four times
Example : In a shooting game, each shooter may shoot four times. Suppose if no bullet hit the target, then the shooter score is zero; if one bullet hit, the score is 15; if two bullets hit, the score is 30; if three bullets hit, the score is 55; and if all bullets hit, the score is 100. The probability of the shooter hitting the target is 0.6, calculate his expected score.

26. Solution: let X denote his score, then the probability function is15 30 55
100 P
EX=44.52

27. The Expected Values Of A Function
Definition: Let X be a discrete rv with set of possible values D and pmf p(x). the expected values or mean value of any function h(X), denoted by E[h(X)] or , is

28. X x1 x2 …… xn p p1 p2 pn
h(X)
h(x1)
h(x2) ……
h(xn) p p1 p2 pn

29. Rules Of Expected Value
Proposition

30. The Variance Of X
Definition: Let X have pmf p(x) and expected value . Then the variance of X, denoted by V(X) or ,or
just , is
The standard deviation (SD) of X is

31. A Shortcut Formula For
Proposition
Rules Of Variance
Proposition

32. 3.4 the Binomial probability Distribution
Properties of a Binomial experiment
1.The experiment consists of a sequence of n identical trials.
2.Two outcomes are possible on each trial. We refer to one as a success (S) and the other as a failure (F).
3.The probability of a success, denoted by p, does not change from trial to trial. Consequently, the probability of a failure, denoted by 1-p,does not change from trial to trial.
4.The trials are independent. (Trials are independent from one to another )

33. Example 1 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.
Definition: An experiment for which conditions 1-4 are satisfied is called a binomial experiment.
Many experiments involve a sequence of independent trials for which there are more than two possible outcomes on any one trial. A binomial experiment can be created by dividing the possible outcomes into two groups.

34. Rule : suppose each trial of an experiment can result in S or F, but the sampling is without replacement from a population of size N. If the sample size (number of trials) n is at most 5% of the population size, the experiment can be analyzed as though it were exactly a binomial experiment.

35. X= the number of S’s among the n trials
The Binomial Random Variable And Distribution
Definition: Given a binomial experiment consisting of n trials, the binomial random variable X associated with this experiment is defined as
X= the number of S’s among the n trials
The binomial distribution is the distribution of number of successes X in n independent success/fail experiments with success probability p. We denote the distribution as Bin(n,p).
If X follows Bin(n, p) distribution, the possible values of X are 0, 1, 2, ..., n. If k is any one of these values,

36. Notation: Because the pmf of a binomial rv X depends on the two parameters n and p, we denote the pmf by b(x;n,p)

37. Example: Each of six randomly selected cola drinkers is given a glass containing cola S and one containing cola F. the glasses are identical in appearance except for a code on the bottom to identify the cola. Suppose there is actually no tendency among cola drinkers to prefer one cola to the other. then p=P(a selected individual prefers S)=.5, so with X=the number among the six who refer S , X~Bin(6,.5)
thus
The probability that at least three prefer S is
The probability that at most one prefer S is

38. Using Binomial Tables
Notation : For X~Bin(n,p), the cdf will be denoted by
Example : Suppose that 20% of all copies of a particular textbook fail a certain binding strength test. Let X denote the number among 15 randomly selected copies that fail the test. Then X has a binomial distribution with n=15 and p=0.2.
1. The probability that at most 8 fail the test is

39. 2. The probability that exactly 8 fail is
3. The probability that at least 8 fail is
4. The probability that between 4 and 7 fail is

40. If we repeat a Bernoulli trial n times and record the total number of successes in these n independent and identical trials, then the total number of successes follows the Binomial distribution.
Let Y = X1 + X Xn where X1, X2, ..., Xn are independent Bernoulli trials with probability of success = p, then Y ~ Bin (n, p)

41. The probability distribution is characterized by two parameters, the sample size n and the probability of success p.
Let's familiarize the Binomial distribution through some examples:

42. It can be seen that the larger is the sample size n, the smoother is the curve. When n is large enough, it is good enough to be proximated by a Normal distribution.

43. E(I) = P(I=1)(1) + P(I=0)(0) = p×1 + q×0 = p
The mean and variance of X
If X ~Bin(n,p)
Mean E(X) = np
In each Bernoulli trial, there are only 2 possible outcomes, success and failure. Suppose we use an indicator I to represent the number of success in each trial. If it is a success, then we have got 1 success; if it is a failure, then we have got 0 success. Then the expected number of success in each trial is
E(I) = P(I=1)(1) + P(I=0)(0) = p×1 + q×0 = p

44. E(X) = E(I) + E(I) + ... + E(I) = p + p + ... + p = np
E(X) is the expected number of successes in n independent trials. It can be simply computed as the sum of the expected success in each trial over n trials,
E(X) = E(I) + E(I) E(I) = p + p p = np

45. Variance Var(X) = np(1-p) = npq
Similary, the variance of the indicator I is
hence the variance of X, the number of successes in n trials, is the sum of n independent variance of I,
Var(X) = Var(I) + Var(I) Var(I) = pq + pq pq = npq = np(1-p)

46. We can also compute the expectation and variance of X theoretically
We can also compute the expectation and variance of X theoretically. As X has the probability mass function,
So the expectation of X is

47. And the variance of X is
where

48. Combining we have

49. 3.6 The Poisson Probability Distribution
Definition:
A discrete random variable is said to follow a Poisson distribution if its probability mass function is in form of
where    > 0.
The Poisson variable is denoted by X ~ P(  ), where   is the mean of the variable X.
The value of is frequently a rate per unit time or per unit area.

50. Example let X denote the number of creatures of a particular type captured in a trap during a given time period . Suppose that X has a Poisson distribution with =4.5, so on average traps will contain 4.5 creatures.
The probability that a trap contains exactly five creatures is
The probability that a trap has at most five creatures is

51. Example
Poisson Distribution