In a game of heads and tails with a fair coin - you win $\$1$ if heads; lose $\$1$ if tails - what is the probability of being $\$5$ up after $25$ tosses?
Suppose that $X_1,X_2,\ldots,X_n$ is a random sample from a distribution with probability function:  $$p_X(x)=\begin{cases} 1/2 &amp;x=-1,1 \\ 0 &amp; \text{otherwise} \end{cases}$$  Now if we define the average   $\bar{X}= \frac{1}{n} \sum X_i$, I need to show that that for an odd $n$ the probability function for $\bar{X}$ is:  $$p_{\bar{X}} (x)= \frac{\binom{n}{\frac{n}{2} (x+1)}}{2^n}$$ for $x=\pm 1/n,\pm 3/n,\ldots,\pm 1$, $0$ otherwise.    Since $P[\sum X_i=k] =P [\bar{X}=k/n ]$, I thought it would be easier to first derive the probability function for the sum. Since they are still $2^n$ n-tuples and each one is equilikely, I need to count the ones whose sum is $k$ for:  $k=\pm 1,\pm 3,\ldots \pm n-2,\pm n$ (since even numbers are not a feasible combination)  The problem here though is that I do not immediately recognize how I could count all these tuples. Could you please help me with this? If it is easier to proceed another way, I am of course all ears.
Iterating each character in a string using Python Primes congruent to 1 mod 6 I came across a claim that I found interesting, but can't seem to prove for some reason. I have the feeling it should be easy  a prime $p$ can be written in the form $p = a^2 -ab +b^2$ for some $a,b\in\mathbb{Z}$ if and only if $p\equiv 1\bmod{6}$ Prove $-(-a)=a$ using only ordered field axioms Including all the jars in a directory within the Java classpath Why should the "H" option not be used in floats? Explain why $E(X) = \int_0^\infty (1-F_X (t)) \, dt$ for every nonnegative random variable $X$ What is potential energy truly? How do I manage (add/remove) all the startup applications in Ubuntu 11.10 and up?  Startup Applications doesn't show all the applications which are started during boot. Can US citizens go to Cuba freely and vice versa after the recent diplomatic relation restoration? A recent full diplomatic relation restoration between the US and Cuba took place, does that mean that US citizens can go to Cuba freely and vice versa? Access-Control-Allow-Origin Multiple Origin Domains? Pagination returns 404 after page 20 I got a paged loop that pulls in 4 posts per page from custom post type. It works great, with the only downside that the pagination wont go over the page/20 everything after that is 404, and the posts_nav_link won't show the next arrow at all. Like the pages just aren't there.  In total there are 119 published posts in the custom post type. If I change posts_per_page to, say 6, then I get access to more of the posts from the end, but it also stops at the page/20.  On the production site the pagination stops at 21, I guess because the live database has more entries.  I got the "Blog pages show at most" setting set to 6.  Here's the loop I'm using:  &lt;?php $paged = get_query_var('paged') ? get_query_var('paged') : 1; $args  = array(     'posts_per_page' =&gt; 4,     'orderby'        =&gt; 'modified',     'post_type'      =&gt; "projekte",     'paged'          =&gt; $paged ); $custom_query = new WP_Query($args); $c = 0; while($custom_query-&gt;have_posts()) :     $custom_query-&gt;the_post();     ?&gt;     &lt;article&gt;...&lt;/article&gt;     &lt;?php endwhile; wp_reset_postdata(); // reset the query ?&gt;   I've gone through the functions.php a million times looking for a some limiting get_pre_posts filter but there's nothing I can see.  I also got an error log setup and nothing is returned there, I've tested it that it works.  Any suggestions? Difference between generalized linear models & generalized linear mixed models How can change in cost function be positive? In  it says     To make gradient descent work correctly, we need to choose the learning rate η to be small enough that Equation (9) is a good approximation. If we don't, we might end up with $\Delta C&gt;0$, which obviously would not be good! At the same time, we don't want $\eta$ to be too small, since that will make the changes $\Delta v$ tiny, and thus the gradient descent algorithm will work very slowly. In practical implementations, $\eta$ is often varied so that Equation (9) remains a good approximation, but the algorithm isn't too slow. We'll see later how this works.   But just a few paragraphs before we established that $\Delta C\approx−\eta\nabla C⋅\nabla C=−\eta\|\nabla C\|^2$ is obviously always negative (for positive $\eta$). So how can $\Delta C$ be positive if we don't choose a small enough learning rate? What is meant there? Proving the AM-GM inequality for 2 numbers $\sqrt{xy}\le\frac{x+y}2$ I am having trouble with this problem from my latest homework.  Prove the arithmetic-geometric mean inequality. That is, for two positive real numbers $x,y$, we have $$ \sqrt{xy}≤ \frac{x+y}{2} .$$ Furthermore, equality occurs if and only if $x = y$.  Any and all help would be appreciated. Page returns 404 with POST variables, but not without I have a problem with my page. I'm trying to make a "Contact Us" page. It's working perfectly fine design-wise. But when I try to send the form to the same page as origin. It returns 404 error.  It only happens when I'm sending the form. I've tried both with GET and POST requests. Nothing seems to work.  I've even commented out the PHPMailer part. It's caused entirely by the extra variables.  If it helps anything, they're "name", "email" and "message". What color would you see if you place 2 mirrors in opposit when one is a one way mirror This image from wikipedia, explains that there occurs a potential drop across a , and an electric field confined to the depletion region.  I already know the reason for the existence of this drop and the calculation of the difference but I have two questions regarding this drop.    1) If the  regions are externally connected using a perfectly conducting wire, why will not any current flow? Connecting the two regions should equalize their potentials (since wire is a resistance less) conductor, and therefore the gradient at the junction is destroyed resulting in a  which is obviously against conservation of energy as the semiconductor has non-zero resistivity. Where will extra potential drops will be created so that Kirchoff's voltage rule holds without any current and the built-in potential difference $V_o$ of the juction persists?    2) (probably naive) If an external Bias is applied, of say $|V|&lt;|V_o|$, then the pn potential difference across the junction will just reduce by that amount ($|V|$). Assuming that the external voltage source is ideal without any resistance, what will be the potential drops which would sum to zero in this case? Will drop due to the resistance of semiconductor play a par in it? If yes, then there cannot be any ideal resistance-less semiconductor junction, can there? Vertical line only in one part of a table I need some help on  putting a vertical line in the second part of the table, I don't know how to do it since the tabularx I declared does need to have that only in the second part. The code I used is:  \begin{tabularx}{\linewidth}{|XX|} \hline \multicolumn{2}{|c|}{\textbf{title name}}\\ \hline something: &amp; else \\ something: &amp; else \\ something: &amp; else \\ \hline \multicolumn{2}{|c|}{bla bla blah}\\ \hline title text: &amp; description 1 \\ title text: &amp; description 1 \\ \hline \end{tabularx}   And the render of the table looks like this:    I just need a \vline between title text and description (it I use \vline it works only for the first line but the description text and sometimes the title, are few lines long). let $x,y,z,w&gt;0$,show that $$(x+y)^3+(y+z)^3+(z+w)^3+(w+x)^3\ge 8(x^2y+y^2z+z^2w+w^2x)$$  it seem use AM-GM inequality to solve it,But I can't it,Thanks Can we access class properties dynamically with Object class? Is it possible to access Apex class properties dynamically? While we can create Object's, and assign any other class to an Object, I'm not sure how I can access the properties of these Object's. Can anyone provide some documentation or references to code which would allow me to do this?   Example:  public class XYZ {    public string s1 { get; set; }    public string s2 { get; set; } }  XYZ testXyzObj = new XYZ(); Object obj = testXyzObj;  // Method does not exist or incorrect signature: [Object].get(String) String s1 = obj.get('s1');