Mixing
Approximating probability of Cumulative Sum of random variables samples from Uniform[-1,1]
$begingroup$
Here is the question:
For $n = 25$ and $50$, approximate the probability $P(max_{1 leq k leq n} S_k > 2sqrt n)$ when the sample observations are iid $U[−1, 1]$, where $S_k= X_1 + X_2+...+X_k$.
I understand that it wants us to approximate the probability that the maximum cumulative sum of the $X_i$s is greater than $2sqrt n$. So, I created a code in R to run the simulation 10000 times and the probability turns out to be very small (i.e 5e-04, 3e-04).
However, I am stuck on how this can be shown by hand. Any help is greatly appreciated!
I also found this theorem in my book that might be related but I am not sure because it has n going to infinity:
$$lim_{nrightarrow infty} P(max_{1 leq k leq n} S_k leq xsqrt n) = G(x), text{ where } G(x)= 2Phi(x)-1. $$
probability statistics probability-distributions statistical-inference
edited Feb 3 at 9:02
pointguard0
1,57611122
asked Feb 3 at 2:45
MBRMBR
414
$endgroup$
add a comment |
$begingroup$
Here is the question:
For $n = 25$ and $50$, approximate the probability $P(max_{1 leq k leq n} S_k > 2sqrt n)$ when the sample observations are iid $U[−1, 1]$, where $S_k= X_1 + X_2+...+X_k$.
I understand that it wants us to approximate the probability that the maximum cumulative sum of the $X_i$s is greater than $2sqrt n$. So, I created a code in R to run the simulation 10000 times and the probability turns out to be very small (i.e 5e-04, 3e-04).
However, I am stuck on how this can be shown by hand. Any help is greatly appreciated!
I also found this theorem in my book that might be related but I am not sure because it has n going to infinity:
$$lim_{nrightarrow infty} P(max_{1 leq k leq n} S_k leq xsqrt n) = G(x), text{ where } G(x)= 2Phi(x)-1. $$
probability statistics probability-distributions statistical-inference
edited Feb 3 at 9:02
pointguard0
1,57611122
asked Feb 3 at 2:45
MBRMBR
414
$endgroup$
add a comment |
$begingroup$
Here is the question:
For $n = 25$ and $50$, approximate the probability $P(max_{1 leq k leq n} S_k > 2sqrt n)$ when the sample observations are iid $U[−1, 1]$, where $S_k= X_1 + X_2+...+X_k$.
I understand that it wants us to approximate the probability that the maximum cumulative sum of the $X_i$s is greater than $2sqrt n$. So, I created a code in R to run the simulation 10000 times and the probability turns out to be very small (i.e 5e-04, 3e-04).
However, I am stuck on how this can be shown by hand. Any help is greatly appreciated!
I also found this theorem in my book that might be related but I am not sure because it has n going to infinity:
$$lim_{nrightarrow infty} P(max_{1 leq k leq n} S_k leq xsqrt n) = G(x), text{ where } G(x)= 2Phi(x)-1. $$
probability statistics probability-distributions statistical-inference
edited Feb 3 at 9:02
pointguard0
1,57611122
asked Feb 3 at 2:45
MBRMBR
414
$endgroup$
Here is the question:
For $n = 25$ and $50$, approximate the probability $P(max_{1 leq k leq n} S_k > 2sqrt n)$ when the sample observations are iid $U[−1, 1]$, where $S_k= X_1 + X_2+...+X_k$.
I understand that it wants us to approximate the probability that the maximum cumulative sum of the $X_i$s is greater than $2sqrt n$. So, I created a code in R to run the simulation 10000 times and the probability turns out to be very small (i.e 5e-04, 3e-04).
However, I am stuck on how this can be shown by hand. Any help is greatly appreciated!
I also found this theorem in my book that might be related but I am not sure because it has n going to infinity:
$$lim_{nrightarrow infty} P(max_{1 leq k leq n} S_k leq xsqrt n) = G(x), text{ where } G(x)= 2Phi(x)-1. $$
probability statistics probability-distributions statistical-inference
probability statistics probability-distributions statistical-inference
edited Feb 3 at 9:02
pointguard0
1,57611122
asked Feb 3 at 2:45
MBRMBR
414
edited Feb 3 at 9:02
pointguard0
1,57611122
asked Feb 3 at 2:45
MBRMBR
414
edited Feb 3 at 9:02
pointguard0
1,57611122
edited Feb 3 at 9:02
pointguard0
1,57611122
edited Feb 3 at 9:02
pointguard0
1,57611122
1,57611122
asked Feb 3 at 2:45
MBRMBR
414
asked Feb 3 at 2:45
MBRMBR
414
asked Feb 3 at 2:45
MBRMBR
414
414
add a comment |
add a comment |
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