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What is what between binomial many flip outcomes and statistical observation?
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I am trying to get my head around normal distribution which evolves as a good approximation for a binomial problem (like coin flips).
Theoretical outcome:
When I have say 50 flips, there is a deterministic no of binomial outcomes (X = no of heads in any outcome sequence), which can be visualized, and graphed. I tried to graph the frequency of X, that is, $n(X_k)$ vs $X_k$, where the $n(X_k)$ are the binomial coefficients. And also the resultant probability $p(X_k)$ vs $X_k$. I also tried to overlay normal approximation for both graphs.
No of flips: 50
p = 0.75 (probability of heads per flip)
Along with applying regular Normal distribution formula on probability curve, I also tried to normal-approximate on frequency curve $n(X_k)$ on LHS as shown below. That is why the term $max(n(X_k))$ multiplied by exponential denoting the red curve which is obviously misplaced regarding which I have questions below.
Statistical outcome:
I also tried to simulate statistically with p(H)=0.75. Since practically we only get one outcome out of $2^50$ possibilities every time, I ran this experiment for say 2000 times. I then collected the $n(X_k)$, and also statistical $p(X_k)$ by just dividing each $n(X_k)$ with total no of outcomes (sum of all $n(X_k)$ ) and plotted them. I also tried to approximate that with normal curves. I get this.
No of flips: 50
No of experiments : 2000 ( 1 experiment $rightarrow$ 50 flips $rightarrow$ 1 output sequence )
p = 0.75
My questions:
- I get vaguely, why $n(X_k)$ in Theoretical outcome still has 25 as mean, because no probability has been associated there yet or inherently we assume equal probability for all? In statistical outcome, you can see $n(X_k)$ has shifted mean to around 37 rightly thus resulting in pdf also on RHS. Theoretical outcome RHS also has pdf sitting on proper mean 37, but LHS looks misplaced.
- If my inference is correct, that by nature, possible outcomes $n(X_k)$ is always probability neutral or independent, how could I normally approximate such theoretical outcomes with varying probability?
- If my inference is wrong, what is missing piece that prevents my binomial outcomes (not their probability which is already in place 37) to shift to mean 37?
- What are proper terminology for both type of probabilities above? Top one is from theoretical outcome, while bottom is statistical. People use one of these to explain normal approximation, so which is correct or better?
On big picture, I am trying to understand how normal distribution evolves from sample distribution and the underlying cause for it.enter preformatted text here
probability probability-distributions normal-distribution binomial-distribution sampling
asked Aug 2 at 7:11
Paari Vendhan
355
add a comment |Â
up vote
0
down vote
favorite
I am trying to get my head around normal distribution which evolves as a good approximation for a binomial problem (like coin flips).
Theoretical outcome:
When I have say 50 flips, there is a deterministic no of binomial outcomes (X = no of heads in any outcome sequence), which can be visualized, and graphed. I tried to graph the frequency of X, that is, $n(X_k)$ vs $X_k$, where the $n(X_k)$ are the binomial coefficients. And also the resultant probability $p(X_k)$ vs $X_k$. I also tried to overlay normal approximation for both graphs.
No of flips: 50
p = 0.75 (probability of heads per flip)
Along with applying regular Normal distribution formula on probability curve, I also tried to normal-approximate on frequency curve $n(X_k)$ on LHS as shown below. That is why the term $max(n(X_k))$ multiplied by exponential denoting the red curve which is obviously misplaced regarding which I have questions below.
Statistical outcome:
I also tried to simulate statistically with p(H)=0.75. Since practically we only get one outcome out of $2^50$ possibilities every time, I ran this experiment for say 2000 times. I then collected the $n(X_k)$, and also statistical $p(X_k)$ by just dividing each $n(X_k)$ with total no of outcomes (sum of all $n(X_k)$ ) and plotted them. I also tried to approximate that with normal curves. I get this.
No of flips: 50
No of experiments : 2000 ( 1 experiment $rightarrow$ 50 flips $rightarrow$ 1 output sequence )
p = 0.75
My questions:
- I get vaguely, why $n(X_k)$ in Theoretical outcome still has 25 as mean, because no probability has been associated there yet or inherently we assume equal probability for all? In statistical outcome, you can see $n(X_k)$ has shifted mean to around 37 rightly thus resulting in pdf also on RHS. Theoretical outcome RHS also has pdf sitting on proper mean 37, but LHS looks misplaced.
- If my inference is correct, that by nature, possible outcomes $n(X_k)$ is always probability neutral or independent, how could I normally approximate such theoretical outcomes with varying probability?
- If my inference is wrong, what is missing piece that prevents my binomial outcomes (not their probability which is already in place 37) to shift to mean 37?
- What are proper terminology for both type of probabilities above? Top one is from theoretical outcome, while bottom is statistical. People use one of these to explain normal approximation, so which is correct or better?
On big picture, I am trying to understand how normal distribution evolves from sample distribution and the underlying cause for it.enter preformatted text here
probability probability-distributions normal-distribution binomial-distribution sampling
asked Aug 2 at 7:11
Paari Vendhan
355
1
It´s not clear what you have done to get the theoretical outcome graphs. What does $max(n(x))...$ mean in this context and why do you use this? And what is $n(X_k)$? Please be more specific and provide a numerical example.
– callculus
Aug 2 at 12:45
Sorry I was not clear enough. $n(X_k)$ is the LHS graph, how many number of $X_k$, per $X_k$. $max(n(X_k))$ is just the maximum number that occurs at mean value, that is when $X_k = 37.5$. I used that function trying to approximate binomial outcomes, but they seem to be misplaced by the mean. I have graphed for numerical problem of No of flips = 50 in theoretical case with probability 0.75 for heads. In statistical case, one experiment is flipping 50 times, and observing no.of heads in resulting sequence. Likewise, I repeated experiment for 2000 times. I have updated now. Is it clearer?
– Paari Vendhan
Aug 2 at 13:36
You asked me to comment on this question, but I'm afraid it's not at all clear to me what you've done, so what I can say is limited. In your upper left graph, it appears that your binomial distribution has $p = 0.5$, but your normal approximation clearly uses $p = 0.75$. There is certainly nothing more interesting to that story than that; the two don't match and shouldn't be expected to match. I don't understand your "My questions" part well enough to be able to comment further. Sorry :-/
– Aaron Montgomery
Aug 4 at 1:45
add a comment |Â
up vote
0
down vote
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up vote
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down vote
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I am trying to get my head around normal distribution which evolves as a good approximation for a binomial problem (like coin flips).
Theoretical outcome:
When I have say 50 flips, there is a deterministic no of binomial outcomes (X = no of heads in any outcome sequence), which can be visualized, and graphed. I tried to graph the frequency of X, that is, $n(X_k)$ vs $X_k$, where the $n(X_k)$ are the binomial coefficients. And also the resultant probability $p(X_k)$ vs $X_k$. I also tried to overlay normal approximation for both graphs.
No of flips: 50
p = 0.75 (probability of heads per flip)
Along with applying regular Normal distribution formula on probability curve, I also tried to normal-approximate on frequency curve $n(X_k)$ on LHS as shown below. That is why the term $max(n(X_k))$ multiplied by exponential denoting the red curve which is obviously misplaced regarding which I have questions below.
Statistical outcome:
I also tried to simulate statistically with p(H)=0.75. Since practically we only get one outcome out of $2^50$ possibilities every time, I ran this experiment for say 2000 times. I then collected the $n(X_k)$, and also statistical $p(X_k)$ by just dividing each $n(X_k)$ with total no of outcomes (sum of all $n(X_k)$ ) and plotted them. I also tried to approximate that with normal curves. I get this.
No of flips: 50
No of experiments : 2000 ( 1 experiment $rightarrow$ 50 flips $rightarrow$ 1 output sequence )
p = 0.75
My questions:
- I get vaguely, why $n(X_k)$ in Theoretical outcome still has 25 as mean, because no probability has been associated there yet or inherently we assume equal probability for all? In statistical outcome, you can see $n(X_k)$ has shifted mean to around 37 rightly thus resulting in pdf also on RHS. Theoretical outcome RHS also has pdf sitting on proper mean 37, but LHS looks misplaced.
- If my inference is correct, that by nature, possible outcomes $n(X_k)$ is always probability neutral or independent, how could I normally approximate such theoretical outcomes with varying probability?
- If my inference is wrong, what is missing piece that prevents my binomial outcomes (not their probability which is already in place 37) to shift to mean 37?
- What are proper terminology for both type of probabilities above? Top one is from theoretical outcome, while bottom is statistical. People use one of these to explain normal approximation, so which is correct or better?
On big picture, I am trying to understand how normal distribution evolves from sample distribution and the underlying cause for it.enter preformatted text here
probability probability-distributions normal-distribution binomial-distribution sampling
asked Aug 2 at 7:11
Paari Vendhan
355
I am trying to get my head around normal distribution which evolves as a good approximation for a binomial problem (like coin flips).
Theoretical outcome:
When I have say 50 flips, there is a deterministic no of binomial outcomes (X = no of heads in any outcome sequence), which can be visualized, and graphed. I tried to graph the frequency of X, that is, $n(X_k)$ vs $X_k$, where the $n(X_k)$ are the binomial coefficients. And also the resultant probability $p(X_k)$ vs $X_k$. I also tried to overlay normal approximation for both graphs.
No of flips: 50
p = 0.75 (probability of heads per flip)
Along with applying regular Normal distribution formula on probability curve, I also tried to normal-approximate on frequency curve $n(X_k)$ on LHS as shown below. That is why the term $max(n(X_k))$ multiplied by exponential denoting the red curve which is obviously misplaced regarding which I have questions below.
Statistical outcome:
I also tried to simulate statistically with p(H)=0.75. Since practically we only get one outcome out of $2^50$ possibilities every time, I ran this experiment for say 2000 times. I then collected the $n(X_k)$, and also statistical $p(X_k)$ by just dividing each $n(X_k)$ with total no of outcomes (sum of all $n(X_k)$ ) and plotted them. I also tried to approximate that with normal curves. I get this.
No of flips: 50
No of experiments : 2000 ( 1 experiment $rightarrow$ 50 flips $rightarrow$ 1 output sequence )
p = 0.75
My questions:
- I get vaguely, why $n(X_k)$ in Theoretical outcome still has 25 as mean, because no probability has been associated there yet or inherently we assume equal probability for all? In statistical outcome, you can see $n(X_k)$ has shifted mean to around 37 rightly thus resulting in pdf also on RHS. Theoretical outcome RHS also has pdf sitting on proper mean 37, but LHS looks misplaced.
- If my inference is correct, that by nature, possible outcomes $n(X_k)$ is always probability neutral or independent, how could I normally approximate such theoretical outcomes with varying probability?
- If my inference is wrong, what is missing piece that prevents my binomial outcomes (not their probability which is already in place 37) to shift to mean 37?
- What are proper terminology for both type of probabilities above? Top one is from theoretical outcome, while bottom is statistical. People use one of these to explain normal approximation, so which is correct or better?
On big picture, I am trying to understand how normal distribution evolves from sample distribution and the underlying cause for it.enter preformatted text here
probability probability-distributions normal-distribution binomial-distribution sampling
asked Aug 2 at 7:11
Paari Vendhan
355
edited Aug 2 at 13:34
asked Aug 2 at 7:11
Paari Vendhan
355
asked Aug 2 at 7:11
Paari Vendhan
355
asked Aug 2 at 7:11
Paari Vendhan
355
355
1
It´s not clear what you have done to get the theoretical outcome graphs. What does $max(n(x))...$ mean in this context and why do you use this? And what is $n(X_k)$? Please be more specific and provide a numerical example.
– callculus
Aug 2 at 12:45
Sorry I was not clear enough. $n(X_k)$ is the LHS graph, how many number of $X_k$, per $X_k$. $max(n(X_k))$ is just the maximum number that occurs at mean value, that is when $X_k = 37.5$. I used that function trying to approximate binomial outcomes, but they seem to be misplaced by the mean. I have graphed for numerical problem of No of flips = 50 in theoretical case with probability 0.75 for heads. In statistical case, one experiment is flipping 50 times, and observing no.of heads in resulting sequence. Likewise, I repeated experiment for 2000 times. I have updated now. Is it clearer?
– Paari Vendhan
Aug 2 at 13:36
You asked me to comment on this question, but I'm afraid it's not at all clear to me what you've done, so what I can say is limited. In your upper left graph, it appears that your binomial distribution has $p = 0.5$, but your normal approximation clearly uses $p = 0.75$. There is certainly nothing more interesting to that story than that; the two don't match and shouldn't be expected to match. I don't understand your "My questions" part well enough to be able to comment further. Sorry :-/
– Aaron Montgomery
Aug 4 at 1:45
add a comment |Â
1
It´s not clear what you have done to get the theoretical outcome graphs. What does $max(n(x))...$ mean in this context and why do you use this? And what is $n(X_k)$? Please be more specific and provide a numerical example.
– callculus
Aug 2 at 12:45
Sorry I was not clear enough. $n(X_k)$ is the LHS graph, how many number of $X_k$, per $X_k$. $max(n(X_k))$ is just the maximum number that occurs at mean value, that is when $X_k = 37.5$. I used that function trying to approximate binomial outcomes, but they seem to be misplaced by the mean. I have graphed for numerical problem of No of flips = 50 in theoretical case with probability 0.75 for heads. In statistical case, one experiment is flipping 50 times, and observing no.of heads in resulting sequence. Likewise, I repeated experiment for 2000 times. I have updated now. Is it clearer?
– Paari Vendhan
Aug 2 at 13:36
You asked me to comment on this question, but I'm afraid it's not at all clear to me what you've done, so what I can say is limited. In your upper left graph, it appears that your binomial distribution has $p = 0.5$, but your normal approximation clearly uses $p = 0.75$. There is certainly nothing more interesting to that story than that; the two don't match and shouldn't be expected to match. I don't understand your "My questions" part well enough to be able to comment further. Sorry :-/
– Aaron Montgomery
Aug 4 at 1:45
1
1
It´s not clear what you have done to get the theoretical outcome graphs. What does $max(n(x))...$ mean in this context and why do you use this? And what is $n(X_k)$? Please be more specific and provide a numerical example.
– callculus
Aug 2 at 12:45
It´s not clear what you have done to get the theoretical outcome graphs. What does $max(n(x))...$ mean in this context and why do you use this? And what is $n(X_k)$? Please be more specific and provide a numerical example.
– callculus
Aug 2 at 12:45
Sorry I was not clear enough. $n(X_k)$ is the LHS graph, how many number of $X_k$, per $X_k$. $max(n(X_k))$ is just the maximum number that occurs at mean value, that is when $X_k = 37.5$. I used that function trying to approximate binomial outcomes, but they seem to be misplaced by the mean. I have graphed for numerical problem of No of flips = 50 in theoretical case with probability 0.75 for heads. In statistical case, one experiment is flipping 50 times, and observing no.of heads in resulting sequence. Likewise, I repeated experiment for 2000 times. I have updated now. Is it clearer?
– Paari Vendhan
Aug 2 at 13:36
Sorry I was not clear enough. $n(X_k)$ is the LHS graph, how many number of $X_k$, per $X_k$. $max(n(X_k))$ is just the maximum number that occurs at mean value, that is when $X_k = 37.5$. I used that function trying to approximate binomial outcomes, but they seem to be misplaced by the mean. I have graphed for numerical problem of No of flips = 50 in theoretical case with probability 0.75 for heads. In statistical case, one experiment is flipping 50 times, and observing no.of heads in resulting sequence. Likewise, I repeated experiment for 2000 times. I have updated now. Is it clearer?
– Paari Vendhan
Aug 2 at 13:36
You asked me to comment on this question, but I'm afraid it's not at all clear to me what you've done, so what I can say is limited. In your upper left graph, it appears that your binomial distribution has $p = 0.5$, but your normal approximation clearly uses $p = 0.75$. There is certainly nothing more interesting to that story than that; the two don't match and shouldn't be expected to match. I don't understand your "My questions" part well enough to be able to comment further. Sorry :-/
– Aaron Montgomery
Aug 4 at 1:45
You asked me to comment on this question, but I'm afraid it's not at all clear to me what you've done, so what I can say is limited. In your upper left graph, it appears that your binomial distribution has $p = 0.5$, but your normal approximation clearly uses $p = 0.75$. There is certainly nothing more interesting to that story than that; the two don't match and shouldn't be expected to match. I don't understand your "My questions" part well enough to be able to comment further. Sorry :-/
– Aaron Montgomery
Aug 4 at 1:45
add a comment |Â
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1
It´s not clear what you have done to get the theoretical outcome graphs. What does $max(n(x))...$ mean in this context and why do you use this? And what is $n(X_k)$? Please be more specific and provide a numerical example.
– callculus
Aug 2 at 12:45
Sorry I was not clear enough. $n(X_k)$ is the LHS graph, how many number of $X_k$, per $X_k$. $max(n(X_k))$ is just the maximum number that occurs at mean value, that is when $X_k = 37.5$. I used that function trying to approximate binomial outcomes, but they seem to be misplaced by the mean. I have graphed for numerical problem of No of flips = 50 in theoretical case with probability 0.75 for heads. In statistical case, one experiment is flipping 50 times, and observing no.of heads in resulting sequence. Likewise, I repeated experiment for 2000 times. I have updated now. Is it clearer?
– Paari Vendhan
Aug 2 at 13:36
You asked me to comment on this question, but I'm afraid it's not at all clear to me what you've done, so what I can say is limited. In your upper left graph, it appears that your binomial distribution has $p = 0.5$, but your normal approximation clearly uses $p = 0.75$. There is certainly nothing more interesting to that story than that; the two don't match and shouldn't be expected to match. I don't understand your "My questions" part well enough to be able to comment further. Sorry :-/
– Aaron Montgomery
Aug 4 at 1:45