Random Variable

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Random Variable
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Random variable

• A random variable ? is a function (rule) that
  assigns a number to each outcome of a chance
  experiment.
• A function ? acts on the elements of its domain
  (the sample space) associating each element with
  a unique real number. The set of all values
  assigned by the random variable is the range of
  this function. If that set is finite, then the
  random variable is said to be finite discrete. If
  that set is infinite but can be written as a
  sequence, then the random variable is said to be
  infinite discrete. If that set is an interval
  then the random variable is said to be
  continuous.

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Example (1)

• A coin is tossed 3 times. Let the random variable
  ? denotes the number of heads that occur in 3
  tosses.
• 1. List the outcomes of the experiment (Find the
  domain of the function ?)
• Answer
• The outcomes of the experiments are those of the
  sample space
• S HHH, HHT, HTH, THH, HTT, THT, TTH, TTT
• the domain of the function (random variable) ?
• 2. Find the value assigned to each outcome by ?
• See the table on the following slide.

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Value of ? Outcome
3 HHH
2 HHT
2 HTH
2 THH
1 HTT
1 THT
1 TTH
0 TTT
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• 3. Let E be the event containing the outcomes to
  which a value of 2 has been assigned by ?. Find
  E!
• Answer
• E HHT, HTH, THH
• 4. Is ? finite discrete, infinite discrete or
  continuous?
• Answer
• The set of the values assigned by ? is
• 0, 1, 2, 3, which is a finite set, hence the
  random variable is ? finite discrete

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Example (2)

• A coin is tossed repeatedly until a head occurs.
  Let the random variable ? denotes the number coin
  tosses in the experiment
• 1. List the outcomes of the experiment (Find the
  domain of the function ?)
• Answer
• The outcomes of the experiments are those of the
  sample space
• S H, TH, TTH, TTTH, TTTTH, TTTTTH, TTTTTTH,
  .
• the domain of the function (random variable) ?
• 2. Find the value assigned to each outcome by ?
• See the table on the following slide which shows
  some of these values.

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Value of ? Outcome
1 H
2 TH
3 TTH
4 TTTH
5 TTTTH
6 TTTTTH
7 TTTTTTH
.. .Etc etc
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• 3. Is ? finite discrete, infinite discrete or
  continuous?
• Answer
• The set of the values assigned by ? is
• 1, 2, 3, 4, 5, 6, 7, 8, 9, .., which is a
  infinite set, that can written as a sequence,
  hence the random variable is ? is infinite
  discrete

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Example (3)

• A flashlight is turned on until the battery runs
  out. Let the Radom variable ? denote the length
  (time) of the life of the battery. What values
  may ? assume ? Is ? finite discrete, infinite
  discrete or continuous?
• Answer
• The set of possible values is the interval 0,8),
  and hence the random variable ? is continuous.

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Probability Distribution of Random variable
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Probability Distribution of Random variable

• A table showing the probability distribution
  associated with the random variable ( which is
  associated with the experiment) rather than with
  the outcomes (which are related to the random
  variable)

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Example (4)

• A coin is tossed 3 times. Let the random variable
  ? denotes the number of heads that occur in 3
  tosses.
• Show the probability distribution of the random
  variable associated with the experiment.

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Value of ? Outcome
3 HHH
2 HHT
2 HTH
2 THH
1 HTT
1 THT
1 TTH
0 TTT
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probability distribution of the random variable X
P(Xx) Value of ?
1/8 0
3/8 1
3/8 2
1/8 3
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Example (5)

• Two dice are rolled. Let the random variable ?
  denotes the sum of the numbers on the faces that
  fall uppermost.
• Show the probability distribution of the random
  variable X.
• Answer
• The values assumed by the random variable X are
  2, 3, 4, 5, 6,.,12, corresponding to the events
  E2, E3, E4,.,E12. The probabilities associated
  with the random variable X corresponding to 2, 3,
  4, ,12 are the probabilities p(E2), p(E3),,
  p(E4), ., p(E12)..

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Sum ofuppermost numbers Event Ek
(1 ,1) 2
(1,2) , (2 ,1) 3
(1,3) , (2 , 2) , (3 ,1) 4
(1, 4) , (2 , 3) , (3 , 2) , (4 ,1) 5
(1 , 5 ) , (2 , 4) , (3 , 3) , (4 , 2) , (5 ,1) 6
(1, 6) , (2 , 5) , (3,4) , (4 , 3) , (5 , 2) , (6 ,1) 7
(2 , 6) , (3 , 5) , (4 , 4) , (5 ,1) , (6 , 2) 8
(3 , 6) , (4 , 5) , (5 , 4) , (6 , 3) 9
(4 , 6) , (5 , 5) , (6 , 4) 10
(5 , 6) , (6 , 5) 11
(6,6) 12
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Probability Distribution of the Random Variable X
p(Xx) x
1/36 2
2/36 3
3/36 4
4/36 5
5/36 6
6/36 7
5/36 8
4/36 9
3/36 10
2/36 11
1/36 12
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Example (6)

• The following table shows the number of cars
  observed waiting in line at the beginning of 2
  minutes interval from 10.00 am to 12.00 noon at
  the drive-in ATM of QNB branch in Ghrafa and the
  corresponding frequency of occurrence. Show the
  probability distribution table of the random
  variable x denoting the number of cars observed
  waiting in line.

Frequency of Occurrence Cars
2 0
9 1
16 2
12 3
8 4
6 5
4 6
2 7
1 8
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• Dividing the frequency by 60 (the sum of these
  numbers- the ones indicating frequency), we get
  the respective probability associated with random
  variable X.
• Thus
• P(X0) 2/60 1/30 0.03
• P(X1) 9/60 3/20 0.15
• P(X2) 16/60 1/10 0.1
• .etc
• See the opposite table

Frequency of Occurrence p(X x) Cars x
2/60 0.03 0
9/60 0.15 1
16/60 0.27 2
12/60 0.20 3
8/60 0.13 4
6/60 0.10 5
4/60 0.07 6
2/60 0.03 7
1/60 0.02 8
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Bar Charts ( Histograms )

• A bar chart or histogram is a graphical means of
  exhibiting probability distribution of a random
  variable.
• For a given probability distribution, a histogram
  is constructed as follows
• 1. Locate the values of the random variable on
  the number line (x-axis)
• 2. Above each number (value) erect a rectangle of
  width 1 and height equal to the probability
  asociated with that number (value).

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Remarks

• 1. The area of rectangle associated with the
  value of the random variable x
• The probability associated with the value x
  (notice tat the width of the rectangle is 1)
• 2. The probability associated with more than one
  value of the random variable x is given by the
  sum of the areas of the rectangles associated
  with those values.

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Example

• Back to the example of three tosses of the coin

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Value of ? Outcome
3 HHH
2 HHT
2 HTH
2 THH
1 HTT
1 THT
1 TTH
0 TTT
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Probability Distribution
x p(Xx)
0 1/8
1 3/8
2 3/8
3 1/8
Value of ? Outcome
3 HHH
2 HHT
2 HTH
2 THH
1 HTT
1 THT
1 TTH
0 TTT
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3
1
2
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• The event F of obtaining at least two heads
• F HHH, HHT, HTH, THH
• P(F) 4/8 ½
• The same result can be obtained from the
  following reasoning
• F is the event (X2) or (X3)
• The probability of this event is equal to
• p(X2) p(X3)
• the sum of the areas of the rectangles
  associated with the values 2 and 3 of the random
  variable
• 1(3/8) 1(1/8)
• 4/8 ½
• on the following slide, these areas are
  identified (the shaded areas)

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Expected Value of a Random Variable
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Mean (Average)

• The average (mean) of the numbers a1, a2,
• a3,. an is denoted by a and equal to
• (a1a2a3.an ) / n
• Example
• Find the average of the following numbers
• 5, 7, 9, 11, 13
• Answer The average of the given numbers ie equal
  to ( 5791113) / 5 45 / 5 9

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Expected Value of a Random Variable( the average
or the mean of the random variable)

• Let x be a random variable that assumes the
  values x1, x2, x3,. xn with associated
  probabilities p1, p2, p3,. pn respectively.
  Then the expected value of x denoted by E(x) is
  equal to
• x1 p1 x2 p2 x3 p3 xn pn

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• The the expected value of x ( the average or the
  mean of X) is a measure of central tendency of
  the probability distribution associated with X.
• As the number of the trails of an experiment gets
  larger and larger the values of x gets closer and
  closer to the expected value of x.
• Geometric Interpretation
• Consider the histogram of the probability
  distribution associated with the random variable
  X. If a laminate (thin board or sheet) is made of
  this histogram, then the expected value of X
  corresponds to the point on the base of the
  laminate at which the laminate will balance
  perfectly when the point is directly over the a
  fulcrum (balancing object).

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Example (7)

• Let X be the random variable giving the sum of
  the dots on the faces that fall uppermost in the
  two dice rolling experiment. Find the expected
  value E(X) of X.

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Recall the probability distribution of the this
random variable
P(Xx) x
1/36 2
2/36 3
3/36 4
4/36 5
5/36 6
6/36 7
5/36 8
4/36 9
3/36 10
2/36 11
1/36 12
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• Solution
• E(X)
• 2(1/36) 3(2/36) 4(3/36) 5(4/36) 6(5/36)
  7(6/36) 8(5/36) 9(4/36) 10(3/36)
  11(2/36) 12(1/36)
• (26122030424036302212) / 36 252 / 36
  
• 7
• Inspecting the histogram on the next slide, we
  notice that the symmetry of the histogram with
  respect to the vertical line x 7, which is the
  expected value of the random variable X.

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Example (8)

• The occupancy rates with corresponding
  probability of hotels A (Which has 52 rooms) B
  (which has 60 rooms) during the tourist season
  are given by the tables on the next slide. The
  average profit per day for each occupied room is
  QR 200 and QR 180 for hotels A B respectively.
  Find
• 1. The average number of rooms occupied per day
  in each hotel.
• 2. Which hotel generates the higher daily profit.

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Hotel B
Occupancy Rate Probability
0.75 0.35
0.80 0.21
0.85 0.18
0.90 0.15
0.95 0.09
1.00 0.02
Hotel A
Occupancy Rate Probability
0.80 0.19
0.85 0.22
0.90 0.31
0.95 0.23
1.00 0.05
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Steps

• I. For each hotel
• 1. Find the expected value of the random
  variable defined to be the occupancy rate in the
  hotel.
• 2. Multiply that by the number of rooms of the
  hotel to find the average number of rooms
  occupied per day.
• 3. Multiply that by the profit made on each room
  per day to find the hotel daily profit.
• II. Compare the result of step 3., for hotels A
  B.

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Solution

• 1. Let the occupancy rate in Hotels A B be the
  random numbers X Y respectively. Then the
  average daily occupancy rate is given by the
  expected values E(X) and E(Y) of X and Y
  respectively. Thus
• E(X) (0.80)(0.19) (0.85)(0.22) (0.90)(0.31)
  (0.95)(0.23) (1.00)(0.05) 0.8865
• The average number of rooms occupied per day in
  Hotel A
• 0.8865 (52) 46.1 rooms
• E(Y) (0.75)(0.35) (0.80)().21) (0.85)(0.18)
  (0.90)(0.15) (0.95)(0.09) (1.00)(0.02)
  0.8240
• The average number of rooms occupied per day in
  Hotel A
• 0.8240 (60) 49.4 rooms

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• 2. The expected daily profit at hotel A
• (46.1 )(200) 9220
• The expected daily profit at hotel B
• (49.4 )(180) 8890
• ? hotel A generates a higher profit