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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.


1d
comment Help with finding $E((\Phi(aX+b)^2))$ when $X$ is standard normal
$\Phi(ax+b)$ is the probability that a normal random variable with mean $-ba^{-1}$ and variance $a^{-2}$ does not exceed $x$. Thus, your integral can be thought of as the probability that two $N(-ba^{-1}, a^{-2})$ random variables $Y$ and $Z$ do not exceed a standard normal random variable $X$, the three random variables being independent.
1d
answered Is the autocovariance function of a sequence identically zero if the sequence is iid?
1d
comment Simple Conditional Expectation Problem
First work out $P(Y = 2\mid A) = \frac{P(Y=2)}{P(A)}$ and $P(Y=5\mid A) = \frac{P(Y=5)}{P(A)}$. (Hint: the two conditional probabilities should sum to $1$. Then, apply the formula for conditional expectation.
Apr
21
comment Isomorphism between finite fields
If you will look at the question that you cite, you will see that the polynomials in question are $x^3-x+1$ and $x^3-x^2+1$, and if you read my answer there, you will see how to construct the isomorphism. And no, you cannot assert that some elements are the same the way you do.
Apr
21
comment Perfect code and even minimum distance
But with a perfect code, every vector belongs to one and only one Hamming sphere of radius $t$ (whatever $t$ is) centered at a codeword. So which sphere does $z$ belong to? The one centered at codeword $x$ or the one centered at codeword $y$ or to some other codeword $\hat{x}$ that is at distance less than $\frac d2$ from $z$? If you choose the last alternative, try and figure out if the distance between $x$ and $\hat{x}$ and between $y$ and $\hat{x}$ to see if both distances are at least $d$. Look at what the triangle inequality has to say about the matter.
Apr
21
answered Perfect code and even minimum distance
Apr
20
comment puzzle on [13,10,3] perfect Hamming code over $\mathbb F_{3}$
Hint: the perfect Hamming code has the property that every vector in $\mathbb F_3^{13}$ is either a codeword in the code or is at Hamming distance $1$ from some codeword. Every possible result of the $13$ games can be expressed as a vector in $\mathbb F_3^{13}$. Now think!
Apr
20
comment constructing “pseudonoise” sequences other than (2^n)-1? (low cyclical autocorrelation)
@endolith There is only one $0$, and that can be replaced with $\pm 1$ with only a small change in the periodic autocorrelation function. The out-of-phase autocorrelation value is constant, but might be $+1$ instead of $-1$. See Golomb's book.
Apr
20
comment Distribution of ratio of uniform and exponential random variables
@Henry Thanks for the confirmation. I wrote a different answer from yours (which I did upvote) to point out to the OP the implicit assumption of independence and also to give the OP a different approach to the problem. I am very much of the same view that you expressed in your comment to Clarinetist: that it is easier to work with the CDF instead of Jacobians.
Apr
19
answered Distribution of ratio of uniform and exponential random variables
Apr
19
comment Distribution of ratio of uniform and exponential random variables
Well, your method of proceeding is poor, though, regrettably, it is often taught as a recipe that can be followed without thinking too much about what's going on, and so I will not attempt to read through your work to see if there is a mistake. But the final answer should be testable. Since $Y \in (0,\infty)$, $Y/X$ takes on values in $(0,\infty)$ also. Is $2u^{-2}, u > 0$ a valid pdf?
Apr
19
comment Normal distribution exercise
What ideas do you have towards solving this problem?
Apr
19
comment A basic question on convergence in distribution
Look for Slutsky's theorem
Apr
19
comment constructing “pseudonoise” sequences other than (2^n)-1? (low cyclical autocorrelation)
These sequences are called Legendre sequences in the literature (cf. Golomb's 1967 book _Shift Register Sequences).
Apr
19
reviewed Close What would be the problem in mathematics if there are no negative numbers in number line?
Apr
19
reviewed Close Contitional Expectation of Sum of two uniform rv
Apr
19
comment convolution and Fourier series
You should define what you mean by convolution. The usual definition of convolution as $$(f\star g)(x) = \int_{-\infty}^\infty f(x-y)g(y)\,\mathrm dy$$ gives a divergent integral when $f$ and $g$ are periodic functions. Hint: you might want to use $\star$ instead of $*$ to denote convolution because you are going to need $*$ to denote complex conjugation.
Apr
18
comment Prove if $A_1\supset A_2,A_1; A_2\in \Im$ then $\Pr(A_1)>\Pr(A_2)$
Is this a problem (e.g. from a book) that you are solving, or one that you made up yourself? Do you know about events of probability $0$ (other than the empty set) that can exist in some sigma-fields?
Apr
18
revised How to obtain Lagrange interpolation formula from Vandermonde's determinant
corrected typo
Apr
18
comment Convolution of uniform random variables
This looks a lot like homework, and if so, please add the homework or self-study tag. Also, what have you done towards attempting a solution of the problem?