Graphing Birthday Function Announcing the arrival of Valued Associate #679: Cesar Manara Planned maintenance scheduled April 17/18, 2019 at 00:00UTC (8:00pm US/Eastern)Problem with the Birthday ProblemBirthday Problem in ProbabilityExpected Duration of Fair Coin TossReal Life Birthday QuestionBirthday problem: expected birthday collision “size”?Generalized birthday paradoxInteractive problem to demonstrate non-intuitive probability results to a large crowd?How to get the general form of the solution of exercise 5.4-2 of CLRS as showed in wikipedia?Birthday problem: using $^nC_r$.Birthday problem for increasing population size
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Graphing Birthday Function
Announcing the arrival of Valued Associate #679: Cesar Manara
Planned maintenance scheduled April 17/18, 2019 at 00:00UTC (8:00pm US/Eastern)Problem with the Birthday ProblemBirthday Problem in ProbabilityExpected Duration of Fair Coin TossReal Life Birthday QuestionBirthday problem: expected birthday collision “size”?Generalized birthday paradoxInteractive problem to demonstrate non-intuitive probability results to a large crowd?How to get the general form of the solution of exercise 5.4-2 of CLRS as showed in wikipedia?Birthday problem: using $^nC_r$.Birthday problem for increasing population size
$begingroup$
I've been trying to graph the birthday function, but have been consistently unable to achieve the commonly shown result despite using the same equation. While I am aware that a simplified form exists, I would like to use the form of (P(366,x))/(366^x). Attatched here is the result I receive from plugging the same equation into Desmos. Note how the function's y varies wildly with only small changes in x. How can I rectify this? Is this an error on Desmos' part? Thanks. I'd also be open to any suggestions regarding how to manipulate that function to produce a correct form.
probability probability-theory
asked Mar 26 at 10:17
Lysander CoxLysander Cox
11
$endgroup$
add a comment |
$begingroup$
I've been trying to graph the birthday function, but have been consistently unable to achieve the commonly shown result despite using the same equation. While I am aware that a simplified form exists, I would like to use the form of (P(366,x))/(366^x). Attatched here is the result I receive from plugging the same equation into Desmos. Note how the function's y varies wildly with only small changes in x. How can I rectify this? Is this an error on Desmos' part? Thanks. I'd also be open to any suggestions regarding how to manipulate that function to produce a correct form.
probability probability-theory
asked Mar 26 at 10:17
Lysander CoxLysander Cox
11
$endgroup$
$begingroup$
Well, at least I believe you should use $365$ instead of $366$ in the equation (see: Calculating the Probability on the Wikipedia site.)
$endgroup$
– Matti P.
Mar 26 at 10:27
$begingroup$
366 for leap years. 365.25 wouldn't feel clean. Roughly the same either way. Any idea why it's showing up like that, though?
$endgroup$
– Lysander Cox
Mar 26 at 10:28
$begingroup$
The vertical axis is logarithmic on the one plot and arithmetic on the other.
$endgroup$
– kimchi lover
Mar 26 at 10:57
$begingroup$
Yes, but what about the discontinuous jumps in my function not present in the other?
$endgroup$
– Lysander Cox
Mar 26 at 11:48
add a comment |
$begingroup$
I've been trying to graph the birthday function, but have been consistently unable to achieve the commonly shown result despite using the same equation. While I am aware that a simplified form exists, I would like to use the form of (P(366,x))/(366^x). Attatched here is the result I receive from plugging the same equation into Desmos. Note how the function's y varies wildly with only small changes in x. How can I rectify this? Is this an error on Desmos' part? Thanks. I'd also be open to any suggestions regarding how to manipulate that function to produce a correct form.
probability probability-theory
asked Mar 26 at 10:17
Lysander CoxLysander Cox
11
$endgroup$
I've been trying to graph the birthday function, but have been consistently unable to achieve the commonly shown result despite using the same equation. While I am aware that a simplified form exists, I would like to use the form of (P(366,x))/(366^x). Attatched here is the result I receive from plugging the same equation into Desmos. Note how the function's y varies wildly with only small changes in x. How can I rectify this? Is this an error on Desmos' part? Thanks. I'd also be open to any suggestions regarding how to manipulate that function to produce a correct form.
probability probability-theory
probability probability-theory
asked Mar 26 at 10:17
Lysander CoxLysander Cox
11
asked Mar 26 at 10:17
Lysander CoxLysander Cox
11
edited Mar 26 at 10:33
Lysander Cox
asked Mar 26 at 10:17
Lysander CoxLysander Cox
11
asked Mar 26 at 10:17
Lysander CoxLysander Cox
11
asked Mar 26 at 10:17
Lysander CoxLysander Cox
11
11
$begingroup$
Well, at least I believe you should use $365$ instead of $366$ in the equation (see: Calculating the Probability on the Wikipedia site.)
$endgroup$
– Matti P.
Mar 26 at 10:27
$begingroup$
366 for leap years. 365.25 wouldn't feel clean. Roughly the same either way. Any idea why it's showing up like that, though?
$endgroup$
– Lysander Cox
Mar 26 at 10:28
$begingroup$
The vertical axis is logarithmic on the one plot and arithmetic on the other.
$endgroup$
– kimchi lover
Mar 26 at 10:57
$begingroup$
Yes, but what about the discontinuous jumps in my function not present in the other?
$endgroup$
– Lysander Cox
Mar 26 at 11:48
add a comment |
$begingroup$
Well, at least I believe you should use $365$ instead of $366$ in the equation (see: Calculating the Probability on the Wikipedia site.)
$endgroup$
– Matti P.
Mar 26 at 10:27
$begingroup$
366 for leap years. 365.25 wouldn't feel clean. Roughly the same either way. Any idea why it's showing up like that, though?
$endgroup$
– Lysander Cox
Mar 26 at 10:28
$begingroup$
The vertical axis is logarithmic on the one plot and arithmetic on the other.
$endgroup$
– kimchi lover
Mar 26 at 10:57
$begingroup$
Yes, but what about the discontinuous jumps in my function not present in the other?
$endgroup$
– Lysander Cox
Mar 26 at 11:48
$begingroup$
Well, at least I believe you should use $365$ instead of $366$ in the equation (see: Calculating the Probability on the Wikipedia site.)
$endgroup$
– Matti P.
Mar 26 at 10:27
$begingroup$
Well, at least I believe you should use $365$ instead of $366$ in the equation (see: Calculating the Probability on the Wikipedia site.)
$endgroup$
– Matti P.
Mar 26 at 10:27
$begingroup$
366 for leap years. 365.25 wouldn't feel clean. Roughly the same either way. Any idea why it's showing up like that, though?
$endgroup$
– Lysander Cox
Mar 26 at 10:28
$begingroup$
366 for leap years. 365.25 wouldn't feel clean. Roughly the same either way. Any idea why it's showing up like that, though?
$endgroup$
– Lysander Cox
Mar 26 at 10:28
$begingroup$
The vertical axis is logarithmic on the one plot and arithmetic on the other.
$endgroup$
– kimchi lover
Mar 26 at 10:57
$begingroup$
The vertical axis is logarithmic on the one plot and arithmetic on the other.
$endgroup$
– kimchi lover
Mar 26 at 10:57
$begingroup$
Yes, but what about the discontinuous jumps in my function not present in the other?
$endgroup$
– Lysander Cox
Mar 26 at 11:48
$begingroup$
Yes, but what about the discontinuous jumps in my function not present in the other?
$endgroup$
– Lysander Cox
Mar 26 at 11:48
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Your plot is the correct one. $P(366,x)/366^x$ is a discrete function, which Desmos displays as a histogram, hence the jumps. The Wikipedia graph is a smooth extension of the discrete graph. They just do that to make it look prettier. Inherently, the function is discrete, since it only makes sense in the context of the birthday problem when $x$ is a positive integer. It should not be smooth.
If you want a smooth function, then you just have to use the Gamma function in place of any factorials. The Gamma function satisfies $Gamma(x+1)=x!$ whenever $x$ is an integer, yet $Gamma$ is defined for all real numbers, and is smooth. Therefore, $fracGamma(366+1)Gamma(366-x+1)366^x$ will be a smooth version of $P(366,x)/366^x$.
answered Mar 26 at 16:25
Mike EarnestMike Earnest
27.9k22152
$endgroup$
add a comment |
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1 Answer
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1 Answer
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$begingroup$
Your plot is the correct one. $P(366,x)/366^x$ is a discrete function, which Desmos displays as a histogram, hence the jumps. The Wikipedia graph is a smooth extension of the discrete graph. They just do that to make it look prettier. Inherently, the function is discrete, since it only makes sense in the context of the birthday problem when $x$ is a positive integer. It should not be smooth.
If you want a smooth function, then you just have to use the Gamma function in place of any factorials. The Gamma function satisfies $Gamma(x+1)=x!$ whenever $x$ is an integer, yet $Gamma$ is defined for all real numbers, and is smooth. Therefore, $fracGamma(366+1)Gamma(366-x+1)366^x$ will be a smooth version of $P(366,x)/366^x$.
answered Mar 26 at 16:25
Mike EarnestMike Earnest
27.9k22152
$endgroup$
add a comment |
$begingroup$
Your plot is the correct one. $P(366,x)/366^x$ is a discrete function, which Desmos displays as a histogram, hence the jumps. The Wikipedia graph is a smooth extension of the discrete graph. They just do that to make it look prettier. Inherently, the function is discrete, since it only makes sense in the context of the birthday problem when $x$ is a positive integer. It should not be smooth.
If you want a smooth function, then you just have to use the Gamma function in place of any factorials. The Gamma function satisfies $Gamma(x+1)=x!$ whenever $x$ is an integer, yet $Gamma$ is defined for all real numbers, and is smooth. Therefore, $fracGamma(366+1)Gamma(366-x+1)366^x$ will be a smooth version of $P(366,x)/366^x$.
answered Mar 26 at 16:25
Mike EarnestMike Earnest
27.9k22152
$endgroup$
add a comment |
$begingroup$
Your plot is the correct one. $P(366,x)/366^x$ is a discrete function, which Desmos displays as a histogram, hence the jumps. The Wikipedia graph is a smooth extension of the discrete graph. They just do that to make it look prettier. Inherently, the function is discrete, since it only makes sense in the context of the birthday problem when $x$ is a positive integer. It should not be smooth.
If you want a smooth function, then you just have to use the Gamma function in place of any factorials. The Gamma function satisfies $Gamma(x+1)=x!$ whenever $x$ is an integer, yet $Gamma$ is defined for all real numbers, and is smooth. Therefore, $fracGamma(366+1)Gamma(366-x+1)366^x$ will be a smooth version of $P(366,x)/366^x$.
answered Mar 26 at 16:25
Mike EarnestMike Earnest
27.9k22152
$endgroup$
Your plot is the correct one. $P(366,x)/366^x$ is a discrete function, which Desmos displays as a histogram, hence the jumps. The Wikipedia graph is a smooth extension of the discrete graph. They just do that to make it look prettier. Inherently, the function is discrete, since it only makes sense in the context of the birthday problem when $x$ is a positive integer. It should not be smooth.
If you want a smooth function, then you just have to use the Gamma function in place of any factorials. The Gamma function satisfies $Gamma(x+1)=x!$ whenever $x$ is an integer, yet $Gamma$ is defined for all real numbers, and is smooth. Therefore, $fracGamma(366+1)Gamma(366-x+1)366^x$ will be a smooth version of $P(366,x)/366^x$.
answered Mar 26 at 16:25
Mike EarnestMike Earnest
27.9k22152
answered Mar 26 at 16:25
Mike EarnestMike Earnest
27.9k22152
answered Mar 26 at 16:25
Mike EarnestMike Earnest
27.9k22152
answered Mar 26 at 16:25
Mike EarnestMike Earnest
27.9k22152
27.9k22152
add a comment |
add a comment |
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$begingroup$
Well, at least I believe you should use $365$ instead of $366$ in the equation (see: Calculating the Probability on the Wikipedia site.)
$endgroup$
– Matti P.
Mar 26 at 10:27
$begingroup$
366 for leap years. 365.25 wouldn't feel clean. Roughly the same either way. Any idea why it's showing up like that, though?
$endgroup$
– Lysander Cox
Mar 26 at 10:28
$begingroup$
The vertical axis is logarithmic on the one plot and arithmetic on the other.
$endgroup$
– kimchi lover
Mar 26 at 10:57
$begingroup$
Yes, but what about the discontinuous jumps in my function not present in the other?
$endgroup$
– Lysander Cox
Mar 26 at 11:48