Lektsii (1) / Lecture 9
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ICEF, 2012/2013 STATISTICS 1 year LECTURES
Lecture 9 |
06.11.2012 |
Combinations
Let’s consider the simplest situation when the outcome space S is a finite set, and all elementary outcomes are equally likely. Assume that S consists of N elementary outcomes s1,..., sN ,
N = # S . Then all elementary outcomes have the same probability N1 . If the number of elements of an event A is NA , # A = NA , then
Pr(A) = NNA .
In other words, in order to find the probability of an event you should calculate the amount of elementary outcomes of an event under consideration.
Example. Two cards are taken successively from the 52 card deck. What is the probability that the first card is red and the second is black? We assume that all combinations of two cards are equally likely. The total number of combinations is N =52 51 (there are 52 ways to take the first card and 51 ways to take the second). Similarly NA = 26 26 . Finally
Pr(A) = 2652 2651 = 0.255 . Note that in this example the order of cards is important: the
combinations (king heart, two of diamonds) and (two of diamonds, king heart) are different combinations.
Ordered and Non-ordered samples (упорядоченные и неупорядоченные выборки)
1. Ordered sample without replacement. There are n balls in a box numbered by 1,…, n, and m balls are selected in a series (последовательно) without replacement (of course, m ≤ n ). It means that we consider the ordered sample. For example, if n =10, m =3, then the samples
(2, 10, 7) and (7, 2, 10) are the different samples. Any sample may be described by the series (a1, a2 ,..., am ) where ai is the number of ith selected ball. Since the sample is made without
replacement it must be a1 ≠ a2 ≠... ≠ am . The total number of such samples is n(n −1)...(n −m +1) .
For m = n we get n! and this number is thetotal number of permutations of n different objects.
2. Non-ordered sample without replacement. Let’s take from the box m balls simultaneously not taking into account the order of selection. In this case two samples are different if one sample contains at least one ball that is not contained in the second sample. Let’s denote the total
n
number of such non-ordered samples by m . Let’s take one such sample. Then if make the
permutations of the elements of this sample we get m! different ordered samples. In other words, each non-ordered sample generates m! different ordered samples. Then
n
m! = n(n −1)...(n −m +1)
m
and
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n! |
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m!(n −m)! |
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m! |
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From (*) it clearly follows that
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n −m |
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Example. A fair coin is tossed 10 times. How many different outcomes of this experiment giving totally three “heads”? The result of any 10 tossing may be described by the sequence
(c1,c2 ,...,c10 ) where ci =1 if ith tossing shows “head”, and ci = 0 if “tail”. For example the realization (0,0,1,1,0,1,0,1,0,0) means that “head’ appears at 3d, 4th, 6th and 8th tossing. Thus the
total number of outcomes with three “heads” coincides with the number of selection of three “places” among ten “places” where it will be “head” (on the remaining “places” there will be
10 |
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10 9 8 |
=120 . It is easy to understand |
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automatically “tail”). But this number is equal to |
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1 2 3 |
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that the total number of different outcomes of the experiment is 210 . Thus if the coin is a fair coin then Pr(three heads) =120210 = 0.117 .
RANDOM VARIABLES
Definition. Random variable is some numerical characteristics which value depends on random outcome.
There are discrete random variables and continuous random variables.
Examples of discrete random variables
1)total sum of dots on two dice;
2)the number of voting for democrats in the sample of 1500 people;
3)number of defective details in the certain sample, etc.
Examples of continuous random variables
1)waiting time of a bus;
2)the result of measurements;
3)output of some enterprise;
4)a weight of a randomly selected men
DISCRETE RANDOM VARIAVBLES, THEIR DISTRIBUTIONS
By definition the set of values of discrete random variable is finite or countable set. In other words if X is a discrete random variable then its values are numbers x1, x2 ,..., xn ,... (this series
may finite or infinite.
Definition. The collection of numbers pi = Pr(X = xi ), i =1,2,..., n,... is called the distribution of
a random variable X.
Obviously any distribution satisfies the following conditions:
1)pi ≥ 0, i =1,2,..., n,... ;
2)∑pi =1 .
i
It is convenient to represent any discrete random variable by means of the following table (in horizontal or vertical orientation):
X |
x1 |
x2 |
… |
xn |
… |
P(X) |
p1 |
p2 |
… |
pn |
… |