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Chapter 4
Random Variables and
Probability Distributions
Mohamed Elhusseiny
[email protected]
Random Variables
• In the previous chapter we dealt with Set Algebra to
represent random experiments and their sample
spaces. In this chapter we will deal with number and
function algebra to consider these experiments and
to consider what is called probability distributions
• A random variable is a function for which each
element of the sample space is influenced by the
function X (random Variable) and assigned a unique
number
X: S
R
Random Variables
X: S
R
• Example: S={ HH, HT, TH, TT}, let X be the
random variable representing the number of heads
that will appear
S={ HH, HT, TH, TT},
X=2
X=1
X=0
X=1
X
0
1
2
Total
P(X)
1/4
2/4
1/4
1
Probability distribution table
Random Variables
• Example:
S={ HHH, HHT, HTH, HTT, THH, THT, TTH,
TTT }, let X be the random variable representing the
number of heads that will appear
{ HHH, HHT, HTH, HTT, THH, THT, TTH, TTT }
X=3
X=1
X=2
X=0
X
0
1
2
3
Total
P(X)
1/8
3/8
3/8
1/8
1
Probability distribution table
Random Variables
• Example: tossing a die twice
S={ (1,1), (1,2), …………………….., (6,5), (6,6)},
let X be the random variable representing the sum of
the two tosses
X
2
3
4
5
6
7
8
9
10
11 12
Total
P(X) 1/ 2/ 3/ 4/ 5/ 6/ 5/ 4/ 3/ 2/ 1/
1
36 36 36 36 36 36 36 36 36 36 36
Probability distribution table
(1,4), (2,3), (3,2), (4,1)
Types of Variables
The variables used by statistics is usually classified
based on there types. There are two types of variables
Variables
Discrete
Continuous
Type of Data
(Qualitative)
Type of Data
(Quantitative)
Nominal Data
Ordinal Data
Numerical Data
Normal Distribution
• Normal Distribution
• The normal distribution is the most important specific
continuous distribution. Given a problem involving a
normal distribution, you should begin by clearly defining
the relevant normal variable X.
• You should next sketch a graph of the normal
distribution and label it with the given information
concerning the mean, standard deviation, and
probabilities.
• Every thing concerning normal distribution family is
transformed into the standard normal random variable Z.
Normal Distribution
Normal Distribution
• the following probabilities, where Z is the standard
normal random variable:
Examples
Examples
Examples
Examples
•
Consider an investment whose profit is normally
distributed with mean of 10% and a standard deviation
of 5%.
Determine the probability of losing money.
ii.
Find the probability that the investment will gain 15%
of the money.
iii.
Find the probability that the investment will gain more
than 15% of the money.
i.
Examples
i.
Let X be a random variable representing the profit
of the investment.
i) P( losing money) = P ( X < 0)
X 
0  10
 P(

)  P ( Z  2)

5
 P ( Z  2)  0.5  P (0  Z  2)
= 0.5 – 0.4772 = 0.0228
Examples
ii. P( gain 15% of the money) = P ( X = 15) = 0
iii. P( gain more than 15% of the money) = P ( X >
15)
 P(
X 
15  10

)  P ( Z  1)
5

 0.5  P (0  Z  1)
= 0.5 – 0.3413 = 0.1587