Random variable transformation with floor function

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Can someone help me with the following question:

For each $ngeq 1$, let $X_n$ be a random variable following an exponential distribution with mean $n$. Determine $F_n$: the distribution function of $X_n - 10 [fracX_n10]$, where $[u]$ denotes the greatest integer lesser or equal than $u$. What is the limit of $F_n$ when $n rightarrow + infty$.

I've tried to calculate that by using Probability Total Law, but since I have the same variable both in $[X_n/10]$ and out the floor function, I was not able to make it.

Using computational simulation I've found that the answer should be a Uniform distribution (0, 10) but I can't prove it mathematically.

probability-distributions uniform-distribution floor-function exponential-distribution

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asked Jul 23 at 13:55

Raul

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Can someone help me with the following question:

For each $ngeq 1$, let $X_n$ be a random variable following an exponential distribution with mean $n$. Determine $F_n$: the distribution function of $X_n - 10 [fracX_n10]$, where $[u]$ denotes the greatest integer lesser or equal than $u$. What is the limit of $F_n$ when $n rightarrow + infty$.

I've tried to calculate that by using Probability Total Law, but since I have the same variable both in $[X_n/10]$ and out the floor function, I was not able to make it.

Using computational simulation I've found that the answer should be a Uniform distribution (0, 10) but I can't prove it mathematically.

probability-distributions uniform-distribution floor-function exponential-distribution

share|cite|improve this question

asked Jul 23 at 13:55

Raul

1

add a comment | 

up vote
0
down vote

favorite

up vote
0
down vote

favorite

Can someone help me with the following question:

For each $ngeq 1$, let $X_n$ be a random variable following an exponential distribution with mean $n$. Determine $F_n$: the distribution function of $X_n - 10 [fracX_n10]$, where $[u]$ denotes the greatest integer lesser or equal than $u$. What is the limit of $F_n$ when $n rightarrow + infty$.

I've tried to calculate that by using Probability Total Law, but since I have the same variable both in $[X_n/10]$ and out the floor function, I was not able to make it.

Using computational simulation I've found that the answer should be a Uniform distribution (0, 10) but I can't prove it mathematically.

probability-distributions uniform-distribution floor-function exponential-distribution

share|cite|improve this question

asked Jul 23 at 13:55

Raul

1

Can someone help me with the following question:

For each $ngeq 1$, let $X_n$ be a random variable following an exponential distribution with mean $n$. Determine $F_n$: the distribution function of $X_n - 10 [fracX_n10]$, where $[u]$ denotes the greatest integer lesser or equal than $u$. What is the limit of $F_n$ when $n rightarrow + infty$.

I've tried to calculate that by using Probability Total Law, but since I have the same variable both in $[X_n/10]$ and out the floor function, I was not able to make it.

Using computational simulation I've found that the answer should be a Uniform distribution (0, 10) but I can't prove it mathematically.

probability-distributions uniform-distribution floor-function exponential-distribution

share|cite|improve this question

asked Jul 23 at 13:55

Raul

1

share|cite|improve this question

share|cite|improve this question

asked Jul 23 at 13:55

Raul

1

asked Jul 23 at 13:55

Raul

1

asked Jul 23 at 13:55

Raul

1

1

add a comment | 

add a comment | 

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$F_n$ can be computed readily. Let $Y_n=X_n-10[fracX_n10]$. Then for $0le y le 10$,
$P(Y_n le y)=sum_k=0^inftyP(10kle X_nle 10k+y)=sum_k=0^infty(e^frac-10kn-e^frac-(10k+y)n)=(1-e^frac-yn)frac11-e^frac-10n$ For large n, $F_n(y)approx fracy10$, which is uniform for $0le yle 10$.

.

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edited Jul 23 at 17:54

answered Jul 23 at 16:58

herb steinberg

93529

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1 Answer
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$F_n$ can be computed readily. Let $Y_n=X_n-10[fracX_n10]$. Then for $0le y le 10$,
$P(Y_n le y)=sum_k=0^inftyP(10kle X_nle 10k+y)=sum_k=0^infty(e^frac-10kn-e^frac-(10k+y)n)=(1-e^frac-yn)frac11-e^frac-10n$ For large n, $F_n(y)approx fracy10$, which is uniform for $0le yle 10$.

.

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edited Jul 23 at 17:54

answered Jul 23 at 16:58

herb steinberg

93529

add a comment | 

up vote
0
down vote

$F_n$ can be computed readily. Let $Y_n=X_n-10[fracX_n10]$. Then for $0le y le 10$,
$P(Y_n le y)=sum_k=0^inftyP(10kle X_nle 10k+y)=sum_k=0^infty(e^frac-10kn-e^frac-(10k+y)n)=(1-e^frac-yn)frac11-e^frac-10n$ For large n, $F_n(y)approx fracy10$, which is uniform for $0le yle 10$.

.

share|cite|improve this answer

edited Jul 23 at 17:54

answered Jul 23 at 16:58

herb steinberg

93529

add a comment | 

up vote
0
down vote

up vote
0
down vote

$F_n$ can be computed readily. Let $Y_n=X_n-10[fracX_n10]$. Then for $0le y le 10$,
$P(Y_n le y)=sum_k=0^inftyP(10kle X_nle 10k+y)=sum_k=0^infty(e^frac-10kn-e^frac-(10k+y)n)=(1-e^frac-yn)frac11-e^frac-10n$ For large n, $F_n(y)approx fracy10$, which is uniform for $0le yle 10$.

.

share|cite|improve this answer

edited Jul 23 at 17:54

answered Jul 23 at 16:58

herb steinberg

93529

$F_n$ can be computed readily. Let $Y_n=X_n-10[fracX_n10]$. Then for $0le y le 10$,
$P(Y_n le y)=sum_k=0^inftyP(10kle X_nle 10k+y)=sum_k=0^infty(e^frac-10kn-e^frac-(10k+y)n)=(1-e^frac-yn)frac11-e^frac-10n$ For large n, $F_n(y)approx fracy10$, which is uniform for $0le yle 10$.

.

share|cite|improve this answer

edited Jul 23 at 17:54

answered Jul 23 at 16:58

herb steinberg

93529

share|cite|improve this answer

share|cite|improve this answer

edited Jul 23 at 17:54

edited Jul 23 at 17:54

edited Jul 23 at 17:54

answered Jul 23 at 16:58

herb steinberg

93529

answered Jul 23 at 16:58

herb steinberg

93529

answered Jul 23 at 16:58

herb steinberg

93529

93529

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add a comment | 

 

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