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I am a retired professor of electrical and computer engineering with a lifetime of experience teaching probability and statistics to undergraduates and error-correcting coding theory to graduate students.


13h
answered Integral of pdf
1d
comment Maximum of Correlated random variables
$$F_Z(z) = P\{Z \leq z\} = P\{X_1\leq z, X_2\leq z, \cdots, X_N\leq z\} = F_{X_1,X_2,\cdots, X_N}(z,z,\cdots, z)$$ where just knowing the correlations between the $X_i$ is not enough to determine the right hand side.
1d
comment Compute the degree of the splitting field
Yes, $X^4+X^3+X^2+X+1$ splits in $\mathbb F_{81}$ (and extensions thereof). You don't need to work too hard at finding "relationships" between the roots of $X^4+X^3+X^2+X+1$. $$X^5-1 = \prod_{i=0}^4(X-\alpha^i) = (X-1)(X^4+X^3+X^2+X+1) \Rightarrow X^4+X^3+X^2+X+1 = \prod_{i=1}^4(X-\alpha^i).$$
1d
comment Compute the degree of the splitting field
The polynomial is a divisor of $X^5-1$ and so its roots are elements of order $5$ in some extension of $\mathbb F_3$. Now, $\mathbb F_{3^n}$ contains a fifth root of unity exactly when $3^n-1$ (the order of the multiplicative group of the field) is divisible by $5$. What is the smallest extension field that contains fifth roots of unity? We need to find the smallest $n$ such that $3^n-1$ is divisible by $5$. Just start from $n=1$ and work your way up....
1d
comment Pooled sample variance, how to prove
See the answer to this recent question on stats.SE for a more general version of this idea.
1d
comment Generator matrix of a Reed-Muller code
Duplicate of this question that was posted earlier this month and has been answered.
2d
comment Compute $\sum_{k=0}^{n}\frac{1}{\binom{n}{k}}$
How about calculating the numerical value of the sum for $n=2,3,4,5$, say, to do at least some work on your question? Or at least telling us why this sum is of interest to you?
2d
revised Is $r_2$ a uniformly at random value in $Z_n$, where $r_2=r_1 . m$
added 48 characters in body
2d
answered Is $r_2$ a uniformly at random value in $Z_n$, where $r_2=r_1 . m$
2d
comment Prove if a|c and b|d and gcd(c,d)=1 then gcd(a,b)=1
Hint: $amx + bry = (mx)a + (ry)b = 1$.
2d
comment Auto-correlation calculation
Asked on stats.SE also.
2d
comment Cumulative distribution function of exponentials
An exponential random variable with mean $\mu$ and standard deviation $\sigma$ does not exist for arbitrary choice of (positive) numbers $\mu$ and $\sigma$; it must be that $(\mu, \sigma) = (\mu, \frac{1}{\sqrt{\mu}})$ for some $\mu > 0$. Otherwise, what you are looking for is a scaled exponential random variable as described on @heropup's answer, and this is something that you have refused to accept previously claiming that you are not interested in a "linear transformation of the parameter"
Oct
21
comment Cumulative distribution function of exponentials
If you know that the CDF of $X$ is $(1-e^{-x})\mathbf 1_{x \in (0,\infty)}$, then you know that $\mu = \sigma = 1$, and even if you are given $n$ independent samples from this distribution, what is there to estimate? The likelihood function is $$\prod_{i=1}^n e^{-x_i} = e^{-\sum_{i=1}^n x_i}$$ which is a constant since the $x_i$ are known quantities (the data), and is of no help in estimating the value of $1$, the known value of the mean $\mu$.
Oct
21
comment Cumulative distribution function of exponentials
If $X$ is an exponential random variable with unknown mean $\mu$, then its CDF is $(1-e^{-x/\mu})\mathbf 1_{x \in (0,\infty)}$, and its pdf (not pmf) is $\frac{1}{\mu}e^{-x/\mu}\mathbf 1_{x \in (0,\infty)}$ and not what you have written. Start from here to get the maximum-likelihood estimate of $\mu$.
Oct
21
comment Is $r_2$ a uniformly at random value in $Z_n$, where $r_2=r_1 . m$
Also asked two hours earlier (and answered in the comments there) on crypto.SE
Oct
21
comment Is $r_2$ a uniformly at random value in $Z_n$, where $r_2=r_1 . m$
Would make any difference if the p and q are very large safe primes? Only if you renege on the promise that $m$ is an arbitrary value in $\mathbb Z_n$ and insist on something simpler like $1 \leq m \leq 20$ or $m = 2^k, k \leq 8$ because this makes life easier for the person implementing the calculation.
Oct
21
comment Is $r_2$ a uniformly at random value in $Z_n$, where $r_2=r_1 . m$
Hint: Since $m$ is arbitrarily chosen, suppose that you perversely chose $m = p$. Try it for the case $p=3, q=5$ which qualify as "large" primes since they are larger than the smallest prime. Is the set $\{3r_1 \bmod 15 \colon r_1 \in \mathbb Z_{15}\}$ the same set as $\{r_1 \colon r_1 \in \mathbb Z_{15}\}$?
Oct
21
comment Use induction and Pascal’s identity to prove that if n > 1, then 1 = n − 1 = n
Have you tried to establish the base case $n=2$? Or do you want someone to write out the complete proof for you?
Oct
20
comment Derivative of integral?
If you want a general method for "writing down" the answer, see the comments following this answer.
Oct
19
comment Covariance for equal variances
Read about the bilinearity of the covariance function. Or, just bull your way through it. Put $W=X+Y, Z=X-Y$ and write $$\operatorname{cov}((W,Z)=E[WZ]-E[W]E[Z]$$ and note that the linearity of expectation allows you to compute $E[W]=E[X+Y]$ and $E[Z] = E[X-Y]$ easily while expanding out $WZ=(X+Y)(X-Y)=X^2-Y^2$ allows you to use linearity of expectation to write $E[WZ]=E[X^2]-E[Y^2]$, etc.