Given a fixed real number $c$, define a random sequence $(X_n)_{n\ge 0}$ by the following random process:
$X_0 = c$ (with probability 1).
For $n>0$, $X_n = U_n X_{n-1}$ where $U_n$ is a real number chosen at random between zero and one, uniformly, and independently of all previous choices $(U_m)_{m
The important part of this problem is understanding the relationship between the fixed real number c, the random variable U, the random process X, and the probability we’re dealing with. The sequence $X_n$ used here is essentially a product of random variables, scaled by a fixed value c.
In this case, the variables $U_n$ are chosen independently and uniformly between 0 and 1. This means that their logarithms, $\log U_n$, are independently and identically distributed (i.i.d.) with a standard uniform distribution.
Now since $X_n = U_n X_{n-1}$, we get that $\log X_n = \log U_n + \log X_{n-1}$. This suggests that the sequence $\{\log X_n\}_{n\ge 0}$ is a random walk. Knowing the properties of log functions and the probability, we can now get the distribution of $\log X_n$ which is essentially the sum of $n$ i.i.d standard uniform random variables.
If we denote by $F_n$ the cumulative distribution function of the sum of $n$ standard uniform random variables, we are given in the first part of the problem that $F_{100}(\log_{10} c)=0.75$. In the second part, we are asked to find $c$ such that $F_{10,000,000}(\log_{10} c)=0.75$.
The important observation here is that the CDF of the sum of $n$ uniformly distributed random variables scales linearly with the number of variables (n). Using this observation, we can deduce that:
$\log_{10} c (100)\times 100,000 = \log_{10} c (10,000,000)$
The given $\log_{10} c (100)\times 100,000 \approx 46.27 \times 100,000 = 4,627,000$
Hence, $\log_{10} c (10,000,000)\approx 4,627,000$. Rounding this to two decimal places gives $\log_{10} c \approx 4,627,000.00$.
This answers the posed question, and it’s always a good idea to verify the conclusion by going through your steps again and making sure everything fits.
But do note that while the logic of distribution and logarithm is correct, the specific magnitude of the resulting $\log_{10} c$ might be overwhelming. In real practice, such large magnitude may raise concerns for possible overflow problems when performing computation in computer programs.
More Answers:
Cube-full Divisors
Random Rectangles
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The important part of this problem is understanding the relationship between the fixed real number c, the random variable U, the random process X, and the probability we’re dealing with. The sequence $X_n$ used here is essentially a product of random variables, scaled by a fixed value c.
In this case, the variables $U_n$ are chosen independently and uniformly between 0 and 1. This means that their logarithms, $\log U_n$, are independently and identically distributed (i.i.d.) with a standard uniform distribution.
Now since $X_n = U_n X_{n-1}$, we get that $\log X_n = \log U_n + \log X_{n-1}$. This suggests that the sequence $\{\log X_n\}_{n\ge 0}$ is a random walk. Knowing the properties of log functions and the probability, we can now get the distribution of $\log X_n$ which is essentially the sum of $n$ i.i.d standard uniform random variables.
If we denote by $F_n$ the cumulative distribution function of the sum of $n$ standard uniform random variables, we are given in the first part of the problem that $F_{100}(\log_{10} c)=0.75$. In the second part, we are asked to find $c$ such that $F_{10,000,000}(\log_{10} c)=0.75$.
The important observation here is that the CDF of the sum of $n$ uniformly distributed random variables scales linearly with the number of variables (n). Using this observation, we can deduce that:
$\log_{10} c (100)\times 100,000 = \log_{10} c (10,000,000)$
The given $\log_{10} c (100)\times 100,000 \approx 46.27 \times 100,000 = 4,627,000$
Hence, $\log_{10} c (10,000,000)\approx 4,627,000$. Rounding this to two decimal places gives $\log_{10} c \approx 4,627,000.00$.
This answers the posed question, and it’s always a good idea to verify the conclusion by going through your steps again and making sure everything fits.
But do note that while the logic of distribution and logarithm is correct, the specific magnitude of the resulting $\log_{10} c$ might be overwhelming. In real practice, such large magnitude may raise concerns for possible overflow problems when performing computation in computer programs.
More Answers:
Cube-full DivisorsRandom Rectangles
Mahjong