Download Chapter 14 Discrete probability distributions

Mathematics and Statistics Chapter 14
Chapter 14 Discrete probability distributions
14.1
Random variable
Random variable is a variable that takes on numerical value and is dependent on chance.
Random variables are denoted by capitals: X, Y, Z,… etc.
The values it takes are denoted by small letters: x, y, z, … or x1, x2, x3, … etc.
Discrete random variable:
The variable can take no more than a countable number of values. E.g.
Continuous random variable:
The variable can take any value in an interval. E.g.
Example 1:
Consider the experiment of tossing a coin three times. The sample space may be written as
S = {HHH, HHT, HTH, HTT, THH,, THT, TTH, TTT}.
If one is concerned only with the number of tails that fall, then a number value of
0, 1, 2 or 3 will be assigned to each sample point, The values may be assumed by some
random variable X, which in this case represents the number of tails when a coin is tossed
three times. We have:
Sample HHH
points
HHT
HTH
HTT
THH
THT
TTH
TTT
X
1
1
2
1
2
2
3
0
Range of X :
14.2
0 X 3
or
X = {0, 1, 2, 3}
Probability distribution:
Probability distribution is a specification (in a form of graph, a table or a function) of the
probability associated with each value of the random variable.
Cont’d example 1:
Many ways to represent the probability distribution:
1. Table
x
0
1
2
3
P(X=x)
1
8
3
8
3
8
1
8
2.
Bar chart
3.
Histogram
4.
Mathematical formula
5.
Others
P.1
Mathematics and Statistics Chapter 14
A function which assigns a probability P(x) to each random variable X = x is called a
probability function or probability distribution.
The two important properties of all discrete probability distributions are
a) 0  P(X = x)  1
b)
 P( X  x) = 1
allx
14.3
Expectation
Expectation (mean or expected value) of a random variable is the center of gravity or the
balancing point of the value x “weighted” by their probability.
Denoted by E(X)(or  or X ) is given by
  E ( X )   xp( x)
all
Consider the example 1, the expectation is:
Example 2:
A company sells custom-made boxes. Suppose these boxes must be ordered in units of 1000,
2000, 4000, 6000 or 12000 and X be the number of boxes sold in each order. Based on past
records, the probability distribution is described as follows.
x
1000
2000
4000
6000
12000
P(X=x)
0.08
.0.27
0.10
0.33
0.22
What is the expected value of X? What is the total number of boxes, in 500 orders, expected
to be sold?
Answer:
Example 3:
A gambler who has 7 dollars plays the following system. At the first toss of a coin, he bets 1
dollar on heads, and quits if he wins. If he loses, he bets 2 dollars on heads at the second
toss, and quits of he wins. If he loses again, he bets his final 4 dollars on heads at the third
toss. Let X be the money gained or lost by the gambler.
(a) Find the probability that he leaves with 8 dollars.
(b) Find the expectation of X.
Answer:
Properties of Expectation
If X, Y, Z, … are random variables on the same sample space S, then
(a)
(b)
(c)
(d)
E(kX) = k E(X),
E(X + k) = E(X) + k,
E(X  Y) = E(X)  E(Y)
E(X  Y  Z  …) = E(X)  E(Y)  E(Z)  …
Cont’d example 3:
Find E(X +1), E(2X), E(X2) and E(X2 + 2X)
P.2
Mathematics and Statistics Chapter 14
14.4
Variance and Standard Deviation
If X is a random variable with the following distribution
x1
x2
x3
…
xn
P (X = x1)
P (X = x2)
P (X = x3)
…
P(X = xn)
Then the variance of X, denoted by Var (X) or  x , is defined as
2
Var ( X ) 
(x  )
2
i
all
P( X  xi )  E[( X   )2 ]
x
where  is the expectation (mean) of X.
The standard deviation is
 s  Var(X )
Cont’d example 2:
Find the variance and the standard deviation of the box order example.
Variance =
Example 4:
A die is thrown, find the variance of the score obtained.
Properties of Variance
(i)
Var (X) =
x
i
all
(ii)
(iii)
(iv)
2
P( X  xi )   2  E ( X 2 )   2 .
x
Var (a) = 0, where a is constant.
Var (X + a) = Var (X).
Var (aX) = a2 Var (X).
(proof: Book2 p.311 & 313)
Ex 14.4 (P.316) #17:
* Standardized random variable Z, which is defined as
Z
X 

and E(Z) = 0, Var(Z) = 1
P.3
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