DEPARTMENT OF QUANTITATIVE METHODS & INFORMATION SYSTEMS Introduction to Business Statistics QM 120 Chapter 5 Discrete random variables and their probability distribution distribution Spring 2008 Dr. Mohammad Zainal
Chapter 5: Random Variables 2 2 Random variables # of PC’s owned Frequency Relative Frequency 0 120 .12 1 180 .18 2 470 .47 3 3 230 230 .23 23 N = 1000 Sum = 1.000 � Let x denote the number of PCs owned by a family. Then x can take any of the four possible values (0, 1, 2, and 3). � A random variable (RV) is a variable whose value is determined by the outcome of a random experiment. y p
Chapter 5: Random Variables 3 3 Discrete random variable � A random variable that assumes countable values is called a discrete random variable. � Examples of discrete RVs: � Number of cars sold at a dealership during a week � Number of houses in a certain block � Number of fish caught on a fishing trip g g p � Number of costumers in a bank at any given day Continuous random variable Continuous random variable � A random variable that can assume any value contained in one or more intervals is called a continuous random variable. one or more intervals is called a continuous random variable.
Chapter 5: Random Variables 4 4 � Examples of continuous RVs : � Height of a person � Height of a person � Time taken to complete a test � Weight of a fish g � Price of a car Example: Classify each of the following RVs as discrete or E l Cl if h f h f ll i RV di continuous. � The number of new accounts opened at a bank during a week Th b f t d t b k d i k � The time taken to run a marathon � The price of a meal in fast food restaurant � The price of a meal in fast food restaurant � The score of a football game � The weight of a parcel
Chapter 5: Probability Distribution of a Discrete RV 5 5 � The probability distribution of a discrete RV lists all the possible bl value l that h the h RV RV can assume and d their h corresponding probabilities. Example: Write the probability distribution of the number of PCs owned by a family. # of PC’s Frequency Relative owned owned Frequency Frequency 0 120 .12 1 180 .18 2 470 .47 3 230 .23 N N = 1000 1000 S Sum = 1.000 1 000
Chapter 5: Probability Distribution of a Discrete RV 6 6 � The following two characteristics must hold for any discrete probability distribution: b b l d b � The probability assigned to each value of a RV x lies in the range 0 to 1; that is 0 ≤ P(x) ≤ 1 for each x. to 1; that is 0 ≤ P(x) ≤ 1 for each x. � The sum of the probabilities assigned to all possible values of x is equal to 1.0; that is Σ P(x) = 1. Example: Each of the following tables lists certain values of x and their probabilities. Determine whether or not each table represents a valid probability distribution. x P(x) x P( x ) x P( x ) 0 .08 0 .25 4 .2 1 .11 1 .34 5 .3 2 2 .39 39 2 2 .28 28 6 6 .6 6 3 .27 3 .13 8 ‐ .1
Chapter 5: Probability Distribution of a Discrete RV 7 7 Example: The following table lists the probability distribution of a discrete RV x. x 0 1 2 3 4 5 6 P( x ) .11 .19 .28 .15 .12 .09 .06 b) P( ≤ 2) b) P(x ≤ 2) c) P(x ≥ 4) ) P( ≥ 4) d) P(1 ≤ x ≤ 4) d) P(1 ≤ ≤ 4) a) P(x = 3) a) P( 3) e) Probability that x assumes a value less than 4 f) Probability that x assumes a value greater than 2 f) Probability that x assumes a value greater than 2 g) Probability that x assumes a value in the interval 2 to 5
Chapter 5: Probability Distribution of a Discrete RV 8 8 Example: For the following table x 1 1 2 2 3 3 4 4 5 5 P( x ) 8 20 24 16 12 a) Construct a probability distribution table. Draw a graph of a) Construct a probability distribution table. Draw a graph of the probability distribution. b) Find the following probabilities iii. P(x ≥ 3) iv. P(2 ≤ x ≤ 4) i. P(x = 3) ii. P(x < 4)
Chapter 5: Mean of a discrete RV 9 9 Mean of a discrete RV � The mean μ ‐ or expected value E(x) ‐ of a discrete RV is the value that you would expect to observe on average if the experiment is repeated again and again i t i t d i d i � It is denoted by = ∑ = ∑ E x E x ( ) ( ) xp x xp x ( ) ( ) � Illustration: Let us toss two fair coins, and let x denote the number of heads observed We should have the following number of heads observed. We should have the following probability distribution table x 0 1 2 P( x ) 1/4 1/2 1/4 Suppose we repeat the experiment a large number of times, say n =4 000 We should expect to have approximately say n =4,000. We should expect to have approximately
Chapter 5: Mean of a discrete RV 10 10 1 thousand zeros, 2 thousand ones, and 1 thousand twos. Then th the average value of x would equal l f ld l + + Sum of measurements 1,000(0) 2,000(1) 1000(2) = 4 000 4,000 n ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 1 1 1 = + + ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ (0) (1) (2) ⎝ ⎝ ⎠ ⎠ ⎝ ⎝ ⎠ ⎠ ⎝ ⎝ ⎠ ⎠ 4 2 4 = ∑ Similarly, if we use the , we would have ( ) ( ) E x xp x = + + E x ( ) 0 (0) 1 (1) P P 2 (2) P = + + 0(1/ 4) 1(1/ 2) 2(1/ 4) = 1
Chapter 5: Mean of a discrete RV 11 11 Example: Recall “ number of PC’s owned by a family” example. Find the mean number of PCs owned by a family. l F d h b f PC d b f l
Chapter 5: Mean of a discrete RV 12 12 Example: In a lottery conducted to benefit the local fire company 8000 tickets are to be sold at $5 each The prize is a company, 8000 tickets are to be sold at $5 each. The prize is a $12,000. If you purchase two tickets, what is your expected gain?
Chapter 5: Mean of a discrete RV 13 13 Example: Determine the annual premium for a $1000 insurance policy covering an event that over a long period of time, has occurred at the rate of 2 times in 100. Let x : the yearly financial gain to the insurance company resulting from the sale of the policy resulting from the sale of the policy C : unknown premium Calculate the value of C such that the expected gain E(x) will Calculate the value of C such that the expected gain E(x) will equal to zero so that the company can add the administrative costs and profit.
Chapter 5: Mean of a discrete RV 14 14 Example: You can insure a $50,000 diamond for its total value by paying a premium of D dollars. If the probability of theft in y p y g p p y a given year is estimated to be .01, what premium should the insurance company charge if it wants the expected gain to equal $1000 .
Chapter 5: Standard Deviation of a Discrete RV 15 15 � The standard deviation of a discrete RV x, denoted by σ , measures the spread of its probability distribution. � A higher value of σ indicates that x can assume values over a larger range about the mean. While, a smaller value indicates that most of the values that can x assume are clustered closely around the mean. � The standard deviation σ can be found using the following Th d d d i i b f d i h f ll i formula: ∑ ∑ σ σ = = − − μ μ 2 2 x p x x p x ( ) ( ) � Hence, the variance σ 2 can be obtained by squaring its standard deviation σ standard deviation σ .
Chapter 5: Standard Deviation of a Discrete RV 16 16 Example: Recall “ number of PC’s owned by a family” example. Find the standard deviation of PCs owned by a family.
Chapter 5: Standard Deviation of a Discrete RV 17 17 Example: An electronic store sells a particular model of a computer notebook. There are only four notebooks in stocks, and the manager wonders what today’s demand for this particular model will be. She learns from marketing department that the probability distribution for x the daily department that the probability distribution for x, the daily demand for the laptop, is as shown in the table. x 1 2 3 4 5 P( x ) .40 .20 .15 .10 .05 Find the mean, variance, and the standard deviation of x. Is it likely that five or more costumers will want to buy a laptop? lik l th t fi t ill t t b l t ?
Chapter 5: Standard Deviation of a Discrete RV 18 18 Example: A farmer will earn a profit of $30,000 in case of heavy rain next year, $60,000 in case of moderate rain, and $15,000 in case of little rain. A meteorologist forecasts that the probability is .35 for heavy rain, .40 for moderate rain, and .25 for little rain next year Let x be the RV that represents next for little rain next year. Let x be the RV that represents next year’s profits in thousands of dollars for this farmer. Write the probability distribution of x. Find the mean, variance, and the standard deviation of x.
Chapter 5: Standard Deviation of a Discrete RV 19 19 Example: An instant lottery ticket costs $2. Out of a total of 10,000 tickets printed for this lottery, 1000 tickets contain a prize of $5 each. 100 tickets contain a prize of $10 each, 5 tickets contain a prize of $1000 each, and 1 ticket has the prize of $5000 Let x be the RV that denotes the net amount player of $5000. Let x be the RV that denotes the net amount player wins by playing this lottery. Write the probability distribution of x. Determine the mean and the standard deviation of x.
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