# Dilip Sarwate

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bio website location Urbana, IL age member for 2 years, 2 months seen 1 hour ago profile views 1,175

I am a retired professor of electrical and computer engineering with a lifetime of experience teaching probability and statistics to reluctant engineering undergraduates.

 Nov21 comment Total probability theorem with normal probability density functions There is no such thing as the random variable $Y|X$; what you are saying is that the conditional density of $Y$ given the value of $X$ is a normal density. Nov11 comment I want a book about Normal distribution and Multivariate normal distribution What is your question? Demands for full explanation are far too broad to be answered here: people have even written whole books on multivariate normal distributions. Nov10 comment How to simulate rare events with extremely low probability Search for "importance sampling" Nov10 comment Finding conditional pdf of $Z$ given $X=x$ where $Z=X+Y$ This question is a duplicate of this one that the OP asked on math.SE about one hour earlier. Nov9 comment Unsure how to calculate mean square error of a variable with a joint distribution $\frac{1}{24}$ is the correct answer. Nov1 comment How can we find the decision boundary for two overlapping continuous uniform distribution? @kunfu If the threshold is $\gamma\in [c,b]$ then an $X$ is mis-classified as a $Y$ with probability $\frac{b-\gamma}{b-a}$ while a $Y$ is mis-classified as an $X$ with probability $\frac{\gamma-c}{d-c}$. Thus, the average probability of mis-classification, being a weighted sum of these two probabilities, is a linear function of $\gamma$ for $\gamma\in [c,b]$. Where do you think its minimum might be? Nov1 comment How can we find the decision boundary for two overlapping continuous uniform distribution? Does $d-c$ equal $b-a$? That is, do the two density functions have the same value where they overlap, or do they have different values? Nov1 comment Does the sum of several distributions become more central or approximated to Normal For starters, read about the Lyapunov version and the Lindeberg version of the CLT on the Wikipedia page on the CLT. Then go read some of the references therein for more in-depth knowledge. Oct28 comment How to estimate a pdf of x under the model of y = x+n, when the pdf of y and the pdf of n are given Do you know if $x$ and $n$ can be treated as independent random variables? If so, the characteristic function $\Psi_y$ of $y$ is the product of the characteristic functions $\Psi_x$ and $\Psi_n$ of $x$ and $n$ respectively, and so, since $\Psi_n$ is nonzero everywhere, you get the characteristic function $\Psi_x$ as being just $\Psi_y/\Psi_n$ from which the pdf of $y$ can be obtained. Oct28 comment Suppose X and Y are each continuous. Are they jointly continuous? I cannot think of a counterexample where x and y would not be jointly continuous. Take $Y = X^2$ where $X \sim U(0,1)$, say, so that $(X,Y)$ is a random point inside the unit square. Oct27 comment what is $\rho$ in the equation $b = \text{Cov}(X,Y)/\text{Var}(X) = \rho \sqrt{{\rm Var}(Y)} / \sqrt{{\rm Var}(X)}$ in regression @NickCox Mathai and Rathie seem to have unusual definitions, different from the most commonly used ones, for various terms in statistics. See, for example, the comments following this answer to a question on essentially the same topic that the OP posted some time ago. Oct27 comment What is a numerical example of $Var(X_1 + X_2) = Var(X_1) + Var(X_2)$ I think this is a bad example in which to include the point which implies that $X$ and $Y$ are uncorrelated; but the converse is false) because for Bernoulli random variable $X$ and $Y$, $E[XY] = E[X]E[Y]$ does imply that $X$ and $Y$ are independent. Oct26 comment Asymptotic normal distribution via the central limit theorem Why is $\hat{p}$ a random variable at all? Also, under almost any model that you might dream up, the CLT will not give you the asymptotic distribution of $\hat{p}$ directly; you have to infer from what the CLT tells you that the asymptotic distribution is that of a degenerate random variable that non-statisticians call a constant. See this answer to a related question. Oct25 comment Finding the expected value of two normal random variables Is this a logical assumption? No, a conditional mean such as $E[b\mid a_1, a_2]$ is not necessarily equal to the unconditional mean $E[b]$. Oct24 comment Maximum Likelihood for shifted Geometric Distribution This looks a lot like a homework problem, and if so, please add the homework or self-study tag. Oct23 comment Joint pdf of a continuous and a discrete rv @cardinal Actually, people with Master's degrees from MIT are themselves likely to be able to provide a completely correct answer to the question, though it would certainly not be satisfactory to Dr Sheldon Cooper under any circumstances. Oct22 comment Joint pdf of a continuous and a discrete rv @DavidMarx As pointed out to you already by cardinal, your answer gives the marginal density of $\min(Y_1,Y_2)$ and this marginal density is a valid pdf as you say, but I too do not see how your answer describes the joint density of $\min(Y_1,Y_2)$ and $V$. Oct22 comment Elementary Probability Questions One basic notion that will go a long way: the sum of conditional probabilities conditioned on different events makes no sense whatever. You will never see $P(A\mid B) + P(A\mid C)$ (but you might see $P(A\mid B)P(B) + P(A\mid C)P(C)$...) Oct22 comment Quadratic lower bound on Gaussian From $f(y)=-y^T\Omega y+k\le p(y)$, doesn't it follow that $f(0) = k \leq p(0)$ and so if you want $p(0) = f(0)$, then it muse be that $k = p(0)$? Why drag in an unnecessary term $k$ to confuse the issue? Oct21 comment Derivation of the Bivariate normal distribution using change-of-variable technique See this document for how the Jacobian method applies to the special case of linear transformations of normal random variables.