Upcoming SlideShare
×

Thanks for flagging this SlideShare!

Oops! An error has occurred.

×
Saving this for later? Get the SlideShare app to save on your phone or tablet. Read anywhere, anytime – even offline.
Standard text messaging rates apply

# Chapter 2 discrete_random_variable_2009

1,238
views

Published on

1 Like
Statistics
Notes
• Full Name
Comment goes here.

Are you sure you want to Yes No
Your message goes here
• Be the first to comment

Views
Total Views
1,238
On Slideshare
0
From Embeds
0
Number of Embeds
0
Actions
Shares
0
59
0
Likes
1
Embeds 0
No embeds

No notes for slide

### Transcript

• 1. Chapter Two Random Variable L1- Discrete random variable L2- Continuous random variable
• 2. Jan 2009 DISCRETE RANDOM VARIABLES
• At the end of the lecture, you will be able to :
• describe types of random variables
• calculate their probability distribution and their cumulative distribution
Learning Objectives:
• 3. Jan 2009
• Random Variable :
• A numerical variable whose measured value can change from is one outcome of random experiment.
• An uppercase letter ( say X ) is used to denote a random variable.
• After the experiment is conducted, the measured value is denoted by a lowercase letter, say x = 10.
• Probability distribution / distribution of a random variable X :
• description of the set of probabilities associated with the possible values of X .
• Probability mass function:
• describe the probability distribution of a discrete random variable
• Probability density function:
• describe the probability distribution of a continuous random variable
• 4. Jan 2009 Examples of random variables: The number of scratches on a surface. Integer values ranging from zero to about 5 are possible values. X = { 0, 1, 2, 3, 4,5} The time taken to complete an examination. Possible values are 15 minutes to over 3 hours. X = { 15 x 180 }
• 5. Jan 2009 DISCRETE RANDOM VARIABLE
• The set x of values of X is finite or countable.
X is a discrete random variable if :
• The Probability Mass Function (pmf) of X is a set of probability values p i assigned to each of the values of x i
• 6. Jan 2009 Example 1: The sample space for a machine breakdown problem is S = { electrical, mechanical, misuse } and each of these failures is associated with a repair cost of about RM200, RM350 and RM50 respectively. Identify the random variable giving reasons for your answer. Example 2: The analysis of the surface of semi conductor wafer records the number of particles of contamination that exceed a certain size. Identify the possible random variable and its values.
• 7. Jan 2009 Probability mass function may typically be given in tabular or graphical form If from Example 1 that P( cost=50)=0.3, P (cost = 200) = 0.2 and P (cost = 350) = 0.5. The probability mass function is given either X = x 50 200 350 f ( x ) = P(X= x ) 0.3 0.2 0.5 tabular form line graph P( x ) 50 200 350 0.3 0.2 0.5 cost
• 8. Jan 2009 CUMULATIVE DISTRIBUTION FUNCTION The cumulative distribution F( x ) of a discrete random variable X with probability mass function f ( x ) is The cumulative distribution of F( x ) is an increasing step function with steps at the values taken by the random variable. The height of the steps are probabilities of taking these values.
• 9. Jan 2009 From Example 1 ( machine breakdowns) : The probability distribution is The following cumulative distribution is obtained X = x 50 200 350 f ( x ) = P(X= x ) 0.3 0.2 0.5
• 10. Jan 2009 Graph of F( x ) F( x ) 50 200 350 0.5 0.3 1.0 Cost ( RM )
• 11. Jan 2009 MEAN AND VARIANCE :- Discrete R.V We can summarize probability distribution by its mean and variance. Mean or expected value is Variance of X is given as Standard deviation of X is  