# Tagged Questions

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### Random Variable Problem with unrestricted Parameters Worded Problem

I have no idea how to go about solving (a) -> (c) For (a) Is K=0.2 because k/1-0.8=1 Hence, P(Z=z) = 0.2(0.8)^x But How do we determine the mean or variance with unrestricted z values.
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### Expectation of Random Variable - Probability Worded Problem

The part I am confused with is (c) I found part (a) which is: p(0) = 7/24, p(1) = 21/24, p(2) = 7/40 and p(3) = 1/120 How do we find the values for a and b, for part (c) ?
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### Expanding the expected value

How to expand: $E(Y+1)^2$ my working out: $E(Y^2)+E(1^2) = E(Y^2)+1$ (I'm not sure why this is though..) Can someone link to or list the rules for expanding the expected value ......
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### Finding values of a constant in a probability distribution

A probability distribution for the random variable $X$ is defined by: $$\mathbb{P}[X=x] = K\cdot(0.9)^x,\quad x = 0,1,2,\ldots$$ It is asked to find $\mathbb{P}[X\geq 2]$. When there is a domain for ...
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### Expectation of uniform distribution with unknown parameter, given maximal (minimal) observation.

Let $x_i \text{ be} ~ i.i.d. ~ \sim Uni[0,\theta]$ $(\theta \text{ unknown})$. Denote $M_n = \max x_i$. So, through circumferential means, I can show that $E(x_1|M_n) = \frac{n+1}{2n} M_n$. The ...
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### Ross probability models questions [on hold]

I am studying for a course and have no professors to talk to live, so I hope some members here can be kind enough to help me. Rather than writing everything out, and splitting it up into different ...
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### Invariance Properties of Brownian Motion

I am trying to make sense of the Scaling-Invariance and Time-Inversion properties of Brownian motion by producing a sample path. For the record, I am using the following definitions. Let $B(t)$ be the ...
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### Proving that three events are mutually independent

Suppose that: the events $A$ and $B\cap C$ are independent. the events $B$ and $A\cap C$ are independent. the events $C$ and $A\cap B$ are independent. the events $A$ and $B\cup C$ ...
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### A counter example of Brownian Motion

Here is an example in my textbook to illustrate why we need the continuous sample path in the definition of Brownian motion. Let $(B_t)$ be a Brownian motion and $U$ be a uniform random variable on ...
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### Proof of “continuity from above” and “continuity from below” from the axioms of probability

One of the consequences of the axioms of probability ($\sigma$ field and probability axiom) is the "infinite subset" and "infinite union" property, I can't figure out how it follows from them. if ...
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### $\mathbb E[\mathbb E(X|Y, Z)|Y]$ or $\mathbb E\{\mathbb E[(X|Y)|Z]\}$?

To begin with, the standard iterated law of probability is as follows. $$\mathbb E X = \mathbb E [\mathbb E(X|Y)]. (1)$$ I am perfectly happy with $(1)$ and there is also some quite good ...
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### Proof of, and requirements for, the reverse of Jensen's Inequality for concave functions

As I understand it, Jensen's Inequality states $$\int_{U}f_{V}\left(h(u)g(u)\right)du\geq f_{V}\left(\int_{U}h(u)g(u)du\right)$$ For a convex function $f_{V}$, a probability distribution $g(u)$ on ...
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### Characteristic functions

Here $E(Y)$ means the expected value of $Y$. 1) Could any one explain for me how to get from (2.7) to (2.8) ? 2) Why does the author know to define $\phi_1(u)$ and $\phi_2(u)$ in such a way? ...
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### Writing probability as log

I have a question regarding the log probability and I am confused on this. The question is: $$\hat P^{(t)}(x)=\sum_{i=1}^N v_i^{(t)}P_i^{(t)}(x)$$ which is some function of size $N$. The question ...
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### Total law of probability in continuous space

I am finding little difficulty in the following definition of total probability specified in a NLP related paper. Say $q^i$ is a partition of my continuous sample space. The authors have defined the ...
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### Unbiased asymptotic variance

Problem: Let $X_1,...,X_n$ be indep. r.v.'s that satisfy, for $i = 1,...,n$, $E(X_i) = \mu_i(\theta)$ & $\mathrm{Var}(X_i)= \sigma_i^2(\theta)$. $\theta$ is the parameter of interest and the ...
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### Existence of density function for a sum of 2 Random Variables

Let's suppose that $Y$ is the normal distribution and that $X$ is another random variable whose density function may or may not exist. Does it follow that $Y+X$ has a density function? I am reading ...