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Continuous Random Variables
L. Wang, Department of StatisticsUniversity of South Carolina;
Slide *
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Uniform distributionSometimes, it is also called rectangular
probability distributionUsed to model random variables that tend to
occur evenly over a range of valuessometimes referred to as the
distribution of little information, because the probability over
any interval of the continuous random variable is the same as for
any other interval of the same width.
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ExampleThe amount of gasoline sold daily at a service station is
uniformly distributed with a minimum of 2,000 gallons and a maximum
of 5,000 gallons. What is the probability that the service station
will sell at least 4,000 gallons?Algebraically: what is P(X 4,000)
?P(X 4,000) = (5,000 4,000) x (1/3000) = .3333
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Waiting time Subway trains on a certain line run every half hour
between mid-night and six in the morning. What is the probability
that a man entering the station at a random time during this period
will have to wait at least 20 minutes.
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Times Between Industrial AccidentsThe times between accidents
for a 10-year period at a DuPont facility can be modeled by the
exponential distribution.where is the accident rate (the expected
number of accidents per day in this case)
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Example of time between accidentsLet Y = the number of days
between two accidents.
Time 12 days 35 days 5 days
Accident Accident Accident #1#2 #3
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Times Between Industrial AccidentsSuppose in a 1000 day period
there were 50 accidents.
or = 50/1000 = 0.05 accidents per day
1/ = 1000/50 = 20 days between accidents
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What is the probability that this facility will go less than 10
days between the next two accidents??f(y) = 0.05e-0.05y
Chart2
0.05
0.0475615
0.0452419
0.0430354
0.0409365
0.03894
0.0370409
0.0352344
0.033516
0.0318814
0.0303265
0.0288475
0.0274406
0.0261023
0.0248293
0.0236183
0.0224664
0.0213707
0.0203285
0.0193371
0.018394
0.0174969
0.0166436
0.0158318
0.0150597
0.0143252
0.0136266
0.012962
0.0123298
0.0117285
0.0111565
0.0106124
0.0100948
0.0096025
0.0091342
0.0086887
0.0082649
0.0078619
0.0074784
0.0071137
0.0067668
0.0064367
0.0061228
0.0058242
0.0055402
0.00527
0.0050129
0.0047685
0.0045359
0.0043147
0.0041042
0.0039041
0.0037137
0.0035326
0.0033603
0.0031964
0.0030405
0.0028922
0.0027512
0.002617
0.0024894
0.0023679
0.0022525
0.0021426
0.0020381
0.0019387
0.0018442
0.0017542
0.0016687
0.0015873
0.0015099
0.0014362
0.0013662
0.0012996
0.0012362
0.0011759
0.0011185
0.001064
0.0010121
0.0009627
0.0009158
0.0008711
0.0008286
0.0007882
0.0007498
0.0007132
0.0006784
0.0006453
0.0006139
0.0005839
0.0005554
0.0005284
0.0005026
0.0004781
0.0004548
0.0004326
0.0004115
0.0003914
0.0003723
0.0003542
0.0003369
y
Y = Time between accidents
f(y)
Probability Density Function
Sheet1
xy
00
10.241971
20.207554
30.15418
40.107982
50.073225
60.048652
70.031873
80.020667
90.013296
100.0085
110.005407
120.003426
130.002163
140.001361
150.000855
160.000535
170.000335
180.000209
190.00013
200.000081
210.00005
220.000031
230.000019
240.000012
250.000007
260.000005
270.000003
280.000002
290.000001
300.000001
310
320
330
340
350
360
370
380
390
400
410
420
430
440
450
460
470
480
490
Sheet1
y
Sheet2
xy
00.05
10.0475615
20.0452419
30.0430354
40.0409365
50.03894
60.0370409
70.0352344
80.033516
90.0318814
100.0303265
110.0288475
120.0274406
130.0261023
140.0248293
150.0236183
160.0224664
170.0213707
180.0203285
190.0193371
200.018394
210.0174969
220.0166436
230.0158318
240.0150597
250.0143252
260.0136266
270.012962
280.0123298
290.0117285
300.0111565
310.0106124
320.0100948
330.0096025
340.0091342
350.0086887
360.0082649
370.0078619
380.0074784
390.0071137
400.0067668
410.0064367
420.0061228
430.0058242
440.0055402
450.00527
460.0050129
470.0047685
480.0045359
490.0043147
500.0041042
510.0039041
520.0037137
530.0035326
540.0033603
550.0031964
560.0030405
570.0028922
580.0027512
590.002617
600.0024894
610.0023679
620.0022525
630.0021426
640.0020381
650.0019387
660.0018442
670.0017542
680.0016687
690.0015873
700.0015099
710.0014362
720.0013662
730.0012996
740.0012362
750.0011759
760.0011185
770.001064
780.0010121
790.0009627
800.0009158
810.0008711
820.0008286
830.0007882
840.0007498
850.0007132
860.0006784
870.0006453
880.0006139
890.0005839
900.0005554
910.0005284
920.0005026
930.0004781
940.0004548
950.0004326
960.0004115
970.0003914
980.0003723
990.0003542
1000.0003369
Sheet2
y
Y = Time between accidents
f(y)
Probability Density Function
Sheet3
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?Recall:
Chart2
0.05
0.0475615
0.0452419
0.0430354
0.0409365
0.03894
0.0370409
0.0352344
0.033516
0.0318814
0.0303265
0.0288475
0.0274406
0.0261023
0.0248293
0.0236183
0.0224664
0.0213707
0.0203285
0.0193371
0.018394
0.0174969
0.0166436
0.0158318
0.0150597
0.0143252
0.0136266
0.012962
0.0123298
0.0117285
0.0111565
0.0106124
0.0100948
0.0096025
0.0091342
0.0086887
0.0082649
0.0078619
0.0074784
0.0071137
0.0067668
0.0064367
0.0061228
0.0058242
0.0055402
0.00527
0.0050129
0.0047685
0.0045359
0.0043147
0.0041042
0.0039041
0.0037137
0.0035326
0.0033603
0.0031964
0.0030405
0.0028922
0.0027512
0.002617
0.0024894
0.0023679
0.0022525
0.0021426
0.0020381
0.0019387
0.0018442
0.0017542
0.0016687
0.0015873
0.0015099
0.0014362
0.0013662
0.0012996
0.0012362
0.0011759
0.0011185
0.001064
0.0010121
0.0009627
0.0009158
0.0008711
0.0008286
0.0007882
0.0007498
0.0007132
0.0006784
0.0006453
0.0006139
0.0005839
0.0005554
0.0005284
0.0005026
0.0004781
0.0004548
0.0004326
0.0004115
0.0003914
0.0003723
0.0003542
0.0003369
y
Y = Time between accidents
f(y)
Probability Density Function
Sheet1
xy
00
10.241971
20.207554
30.15418
40.107982
50.073225
60.048652
70.031873
80.020667
90.013296
100.0085
110.005407
120.003426
130.002163
140.001361
150.000855
160.000535
170.000335
180.000209
190.00013
200.000081
210.00005
220.000031
230.000019
240.000012
250.000007
260.000005
270.000003
280.000002
290.000001
300.000001
310
320
330
340
350
360
370
380
390
400
410
420
430
440
450
460
470
480
490
Sheet1
y
Sheet2
xy
00.05
10.0475615
20.0452419
30.0430354
40.0409365
50.03894
60.0370409
70.0352344
80.033516
90.0318814
100.0303265
110.0288475
120.0274406
130.0261023
140.0248293
150.0236183
160.0224664
170.0213707
180.0203285
190.0193371
200.018394
210.0174969
220.0166436
230.0158318
240.0150597
250.0143252
260.0136266
270.012962
280.0123298
290.0117285
300.0111565
310.0106124
320.0100948
330.0096025
340.0091342
350.0086887
360.0082649
370.0078619
380.0074784
390.0071137
400.0067668
410.0064367
420.0061228
430.0058242
440.0055402
450.00527
460.0050129
470.0047685
480.0045359
490.0043147
500.0041042
510.0039041
520.0037137
530.0035326
540.0033603
550.0031964
560.0030405
570.0028922
580.0027512
590.002617
600.0024894
610.0023679
620.0022525
630.0021426
640.0020381
650.0019387
660.0018442
670.0017542
680.0016687
690.0015873
700.0015099
710.0014362
720.0013662
730.0012996
740.0012362
750.0011759
760.0011185
770.001064
780.0010121
790.0009627
800.0009158
810.0008711
820.0008286
830.0007882
840.0007498
850.0007132
860.0006784
870.0006453
880.0006139
890.0005839
900.0005554
910.0005284
920.0005026
930.0004781
940.0004548
950.0004326
960.0004115
970.0003914
980.0003723
990.0003542
1000.0003369
Sheet2
y
Y = Time between accidents
f(y)
Probability Density Function
Sheet3
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In General
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Exponential Distribution
Chart2
0.05
0.0475615
0.0452419
0.0430354
0.0409365
0.03894
0.0370409
0.0352344
0.033516
0.0318814
0.0303265
0.0288475
0.0274406
0.0261023
0.0248293
0.0236183
0.0224664
0.0213707
0.0203285
0.0193371
0.018394
0.0174969
0.0166436
0.0158318
0.0150597
0.0143252
0.0136266
0.012962
0.0123298
0.0117285
0.0111565
0.0106124
0.0100948
0.0096025
0.0091342
0.0086887
0.0082649
0.0078619
0.0074784
0.0071137
0.0067668
0.0064367
0.0061228
0.0058242
0.0055402
0.00527
0.0050129
0.0047685
0.0045359
0.0043147
0.0041042
0.0039041
0.0037137
0.0035326
0.0033603
0.0031964
0.0030405
0.0028922
0.0027512
0.002617
0.0024894
0.0023679
0.0022525
0.0021426
0.0020381
0.0019387
0.0018442
0.0017542
0.0016687
0.0015873
0.0015099
0.0014362
0.0013662
0.0012996
0.0012362
0.0011759
0.0011185
0.001064
0.0010121
0.0009627
0.0009158
0.0008711
0.0008286
0.0007882
0.0007498
0.0007132
0.0006784
0.0006453
0.0006139
0.0005839
0.0005554
0.0005284
0.0005026
0.0004781
0.0004548
0.0004326
0.0004115
0.0003914
0.0003723
0.0003542
0.0003369
y
Y = Time between accidents
f(y)
Probability Density Function
Sheet1
xy
00
10.241971
20.207554
30.15418
40.107982
50.073225
60.048652
70.031873
80.020667
90.013296
100.0085
110.005407
120.003426
130.002163
140.001361
150.000855
160.000535
170.000335
180.000209
190.00013
200.000081
210.00005
220.000031
230.000019
240.000012
250.000007
260.000005
270.000003
280.000002
290.000001
300.000001
310
320
330
340
350
360
370
380
390
400
410
420
430
440
450
460
470
480
490
Sheet1
y
Sheet2
xy
00.05
10.0475615
20.0452419
30.0430354
40.0409365
50.03894
60.0370409
70.0352344
80.033516
90.0318814
100.0303265
110.0288475
120.0274406
130.0261023
140.0248293
150.0236183
160.0224664
170.0213707
180.0203285
190.0193371
200.018394
210.0174969
220.0166436
230.0158318
240.0150597
250.0143252
260.0136266
270.012962
280.0123298
290.0117285
300.0111565
310.0106124
320.0100948
330.0096025
340.0091342
350.0086887
360.0082649
370.0078619
380.0074784
390.0071137
400.0067668
410.0064367
420.0061228
430.0058242
440.0055402
450.00527
460.0050129
470.0047685
480.0045359
490.0043147
500.0041042
510.0039041
520.0037137
530.0035326
540.0033603
550.0031964
560.0030405
570.0028922
580.0027512
590.002617
600.0024894
610.0023679
620.0022525
630.0021426
640.0020381
650.0019387
660.0018442
670.0017542
680.0016687
690.0015873
700.0015099
710.0014362
720.0013662
730.0012996
740.0012362
750.0011759
760.0011185
770.001064
780.0010121
790.0009627
800.0009158
810.0008711
820.0008286
830.0007882
840.0007498
850.0007132
860.0006784
870.0006453
880.0006139
890.0005839
900.0005554
910.0005284
920.0005026
930.0004781
940.0004548
950.0004326
960.0004115
970.0003914
980.0003723
990.0003542
1000.0003369
Sheet2
y
Y = Time between accidents
f(y)
Probability Density Function
Sheet3
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If the time to failure for an electrical component follows an
exponential distribution with a mean time to failure of 1000 hours,
what is the probability that a randomly chosen component will fail
before 750 hours?Hint: is the failure rate (expected number of
failures per hour).
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Mean and Variance for an Exponential Random VariableNote: Mean =
Standard Deviation
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The time between accidents at a factory follows an exponential
distribution with a historical average of 1 accident every 900
days. What is the probability that that there will be more than
1200 days between the next two accidents?
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If the time between accidents follows an exponential
distribution with a mean of 900 days, what is the probability that
there will be less than 900 days between the next two
accidents?
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Relationship between Exponential & Poisson
DistributionsRecall that the Poisson distribution is used to
compute the probability of a specific number of events occurring in
a particular interval of time or space.Instead of the number of
events being the random variable, consider the time or space
between events as the random variable.
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Relationship between Exponential & Poisson
Exponential distribution models time (or space) between Poisson
events.TIME
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Exponential or Poisson Distribution?We model the number of
industrial accidents occurring in one year.
We model the length of time between two industrial accidents
(assuming an accident occurring is a Poisson event).
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Recall: For a Poisson Distributiony = 0,1,2,where is the mean
number of events per base unit of time or space and t is the number
of base units inspected.The probability that no event occurs in a
span of time (or space) is:
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Now let T = the time (or space) until the next Poisson event.In
other words, the probability that the length of time (or space)
until the next event is greater than some given time (or space), t,
is the same as the probability that no events will occur in time
(or space) t.
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