Probability and Statistics

Where to find extremely tough problems on random vectors?

It's my go to method for dealing with Bayes type questions, whenever I condition over something I first make a statement like Siong Thye Goh did by P( A f, B s|I f) and then I think on how to draw that tree

i just view everything as maths

I guess not. The problem is asking for the probability of at least TWO simultaneous active users, while the answer calculates the probability of at least one active user.

Hi guys, do you think this answer is correct? math.stackexchange.com/a/1289281/624014

Suppose that the time until a bus arrives is modeled as being equally likely to be Exp(1.1), Exp(1.2), and Exp(2). The bus actually arrives are time 0.4. What are the posterior probabilities for each model?

A remark is the notation difference between the formula, the $x_i$ refers to the $i$-th data in the first formula. In the second formula $x_i$ refers to the $i$-th distinct value.

The formula are equal because we can use use multiplication to group the same number. We can group the same number together and count the sum by using multiplication. Example, $44+46+47+47+47+51=44(1)+46(1)+47(3)+51$. @Swarup

Forget fast calculation formulas. There are two basic formulas, one for calculating the mean from unsorted/unordered data (jargon alert), and the other for computing it from sorted/ordered data (jargon alert). I believe they are, respectively: $$ \bar{x} = \frac{1}{n} \sum _{i=1} ^n x_i $$ $$ \bar{x} = \frac{1}{n} \sum _{i=1} ^k f_i x_i $$ I don't even know why both of them equal the same thing.

@GabrielRomon Sheldon Ross' A first course in Probability is my favourite as an introductory text to Probability.

@SangchulLee I was wondering, do you have any recommendations (problem books, textbooks) for the subjects real analysis, measure theory and probability ?

I mean, this is little different from what you usually encounter in calculus class, such as $$ \int_{0}^{1} \int_{0}^{1} x \, dx dy = \int_{0}^{1} \int_{0}^{1} y \, dx dy $$

Let $(X_n; n \geq 1)$ be a collection of independent positive identically distributed random variables, with density $f(x)$. They are inspected in order from $n = 1$. An observer conjectures that $X_1$ will be greater than all the subsequent $X_n, n \geq 2$. Show that this conjecture will be proved wrong with probability 1.

the key word is "know" to what extend can we claim that we know something

Focus on the first term on the right, we have $P(T_{i-1} < \infty |S_0 =i) = Pr(T_0 < \infty | S_0 = 1)$

I am wondering why is the (SS_res)/sigma^2 ~ chi-square (n-p-1) :))

hmm... i heard people like sheldon ross book or bertsekas book. I like David Garmanik's note. here is a website recommending some books though i rarely read link

I am trying to find a good way to study for the final

related

excess life at time $t$ exceeds $x$ means ( I am $t$ age years old now, I get to live for another $x$ years) if and only there are no renewal in the interval $(t, t+x]$ (there is no death within $(t, t+x]$ )

there is a result that if it is irreducible, then it is unique right? if it is not reducible, consider a markov chain with two irreducible components, then each component has their own stationary distribution (setting the other component that corresponds to the other component to be 0). Hence we have construted two stationary distribution right? also any convex combination of them is another stationary distribution.

recall that summing random variable corresponds to convolution

suppose that$ x_n \to x$ in distribution as n $\to$ $\infty$ prove that$ x_n+c $ $\to$ x+c and $cx_n \to cx$ in distribution

Just want to add