A company puts five different types of prizes into their cereal boxes, one in each box and in equal proportions. If a customer decides to collect all five prizes, what is the expected number of boxes of cereals that he or she should buy?  TRY Let $X$ be the number of boxes customer buys. For $i=1,2,3,4,5$, write $$ X_{ij} = \begin{cases} 1, &amp; \text{ith prize is inside jth box} \\ 0, &amp; \text{otherwise} \end{cases} $$ As I understand the problem, the number of boxes is not given so we may write $$ X = \sum_{j=1}^{\infty} \sum_{i=1}^5 X_{ij} $$ So $$ E(X) = \sum_{j \geq 1 } \sum_{i=1}^5 E(X_{ij}) $$ We know $E(X_{ij}) = P(X_{ij}=1)$ so we need to find proobability that ith  prize is inside jth box. Here is the part where I get stuck. Am I appraoching this problem the correct way?
What is the average number of times it would it take to roll a fair 6-sided die and get all numbers on the die?  The order in which the numbers appear does not matter.  I had this questions explained to me by a professor (not math professor), but it was not clear in the explanation.  We were given the answer $(1-(\frac56)^n)^6 = .5$ or $n = 12.152$  Can someone please explain this to me, possibly with a link to a general topic?
How to Create a Half-Turn Staircase? I would like to create a half-turn staircase without a landing. Is there an easy way to accomplish this without having to manually calculate and create each step individually?  Here’s an image similar to what I would like to achive:    I have found the Archimesh add-on including . While I can bend the created stairs, I haven’t found any way to get anywhere close to the image above. My stairs just always look out of shape and I don’t know how to get the outline of the staircase into a square.  I should mention that I only have very little experience with Blender modeling, so it’s well possible that I’m just missing some curve modification basics. Perhaps Archimesh is also the wrong approach for this in the first place? Renting a car in Ireland & Great Britain: 1 rental or 2? My wife, two kids and I will be traveling in Ireland and the UK Nov. 1st through 21st. We're flying into and out of Dublin. Our plan is to do a clockwise tour: Fly into Dublin -> Belfast -> Scotland -> England/London -> Back to Dublin to fly home.  We've been planning to rent a car when we land in Dublin (doing just things outside Dublin at the start of the trip) and have a car all the way until London, then turn in the car and take a train/ferry to get back to Dublin before flying home.  However, reading through the car rental section of Rick Steve's 2012 Ireland book is causing us to question if this is the right plan.  He says (p. 480):     If your trip covers both Ireland and Great Britain, you're better off   with two separate car rentals, rather than paying for your car to ride   the ferry between the two islands. On an all-Ireland trip, you can   drive your rental car from the Republic of Ireland into Northern   Ireland, but be aware of drop-off charges ($75-150) if you return it   in the North.   So it seems like he recommends against doing one car rental for both islands, but he also recommends against renting in the Republic of Ireland and returning it in Northern Ireland (our other option for our current itinerary).  So we're not sure which is best here.  A few questions about this:   If we do one car rental all the way from Dublin to London, will there be a similar $75-$150 fee? Is there an easy place to return a car on the Northern Ireland side of one of the Northern Ireland -> Scotland ferries, and an easy place on the Scotland side to rent a car?  We'd like to minimize the number of buses/whatever we need to take between car rentals since we'll be lugging around car seats for both of our children (on top of carrying them and our luggage). Will one way be significantly cheaper than the other?  Besides the basic cost of taking the car on the ferry and the drop-off charges, our credit card will cover insurance in Scotland/England but not in Ireland, so we could potentially get a better rental rate by renting a second vehicle once we're on Great Britain. Although, I know you often get a better rate when keeping a car rental for longer, so that may negate any benefit from the credit card covering the insurance. Given a continuous random variable $X$ and the pdf find.... There is a pattern excercise:  First pattern: We have $X$ and the probability density function is usually:  $$ f(x) = \begin{cases}   {ax+bx^2} &amp;  {0\lt x \lt 1} \\   0 &amp; \text{else}   \end{cases} $$  Then we are given $E|X| = 0.6$   And usually asks to find:  a) $P\{X&gt;0.5\}$  b) $Var(X)$    Second pattern:  $$ f(x) = \begin{cases}   {cx^4} &amp;  {0\lt x \lt 2} \\   0 &amp; \text{else}   \end{cases} $$  Where c is a constant. Find:  a) Mean value of $X$  b) Variance of $X$     Third pattern (it's more like a reverse problem):  We are given $Y= \sqrt{X}$, find it's cumulative distribution function (CDF) and  it's probability density function (pdf).    Update:  The third one is transferred to a .  This is not homework. I'm self-studying and in need of some help. Thank you. When to use an offset in a Poisson regression? Does anybody know why offset in a Poisson regression is used? What do you achieve by this? A problem with my reasoning in a problem about combinations I was given the following problem to solve:     A committee of five students is to be chosen from six boys and five   girls. Find the number of ways in which the committee can be chosen,   if it includes at least one boy.   My method was $\binom{6}{1}\binom{10}{4}= 1260$, using the logic of choosing $1$ boy, then choosing the rest. This was wrong, as the answer was $\binom{11}{5}-\binom{5}{5}= 461$.  The correct answer's logic was committee with no restrictions – committee with no girls.  Why was my method wrong? Please help... VHDL Transition Function Research statement ideas got used by interviewing committee Can you Start Skyrim 'silent' without the launcher showing up? How can I run a command after boot? I would like to run the simple shell command echo 1 &gt; /proc/sys/kernel/sysrq at each boot, to enable the sysrq keys. When in boot should this be done, and where should I put a script to do it? Proving that two vector spaces of equal dimension are isomorphic Given the following problem     Let $U$ and $V$ be finite dimensional vector spaces, such that $\dim(U) = \dim(V)$. Prove that $U$ and $V$ are isomorphic.   I was just wondering if somebody could critique and validate my proof.  Proof  Two vector spaces $U$ and $V$ are isomorphic if and only if there exists a bijection $\phi: U \to V$. That is, an invertible linear map between the two. Let $\dim(U) = \dim(V) = n$ and let $\left\{u_1, \dots, u_n\right\}$ and $\left\{v_1, \dots, v_n\right\}$ be bases for $U$ and $V$, respectively. It follows that any vector $u \in U$ can be written as  $$ u = c_1u_1 + \cdots + c_nu_n $$  We define $\phi: U \to V$ as  $$ \phi(c_1u_1 + \cdots + c_nu_n) = c_1v_1 + \cdots + c_nv_n $$  Observe that $\phi$ is invertible. That is, given a vector  $$ v = d_1v_1 + \cdots + d_nv_n $$  in $V$, we define $\phi^{-1}: V \to U$ as  $$ \phi^{-1}(d_1v_1 + \cdots + d_nv_n) = d_1u_1 + \cdots + d_nu_n $$  Since there exists a bijection between $U$ and $V$, they are isomorphic. Distribution of $-\log X$ if $X$ is uniform. For $X$ and $Y$ random variables; $X$ follows the uniform distribution.  (1): if $Y=-\log X$  (2): then it can be shown that $-\log X$ is distributed as $\exp(1)$ {i.e. exponential with mean 1}.  Why is this so? Intuitively statement (2) make sense to me. But i'd like a mathematical proof.  -Probably wrong working:  (1) seems to imply $\exp(-Y) = X$ (which is like saying $X$ is exponentially distributed, which is a contradiction, since its actually uniform!); or is it incorrect for me to do this since $X$ and $Y$ are random variables?  Ultimately how do I prove (2)?  Thanks Test coverage for setCookies method I am adding cookie to my page in the controller using  ApexPages.currentPage().setCookies(new Cookie[]{CookieName});  When am trying to get the code coverage, my test class is not covering the above line.  How to get the code coverage for the above setcookie method in the test class. How does the two slit experiment work? Whilst going through 'A brief history of time', I faced difficulty in understanding the . How can an individual electron cause fringes on screen? I was unable to understand it? Please explain it simply. How do I edit my profile? This is probably a noob question, but I forgot how to edit my profile.   Can anyone help me? There is no visible button for me. Why do some Jedi disappear when they die while others don't? Some Jedi disappear when they die:   In Star Wars Episode IV: A New Hope       At the end of his duel with Vader on the first Death Star, Obi-Wan Kenobi let down his guard and concentrated for a moment, disappearing entirely just as Vader's lightsaber passed through his empty robe.   In Star Wars Episode VI: Return of the Jedi      When Luke returned to Dagobah in the Return of the Jedi, Yoda also disappears when he peacefully dies.  In Star Wars Episode VIII: The Last Jedi      When Luke is exhausted after “being” on Crait he looks into the Horizon and disappears.    And others don't:   Star Wars Episode I: The Phantom Menace      When Qui-Gon Jinn's dies after his duel with Darth Maul, he doesn't  disappear and is cremated.  Star Wars Episode VI: Return of the Jedi      When Vader/Anakin Skywalker dies on the second Death Star, he does not disappear and he is cremated by Luke.      Also, they all became a "ghost" after their death. Show that $\frac 1{1+x+y^{-1}}+\frac1{1+y+z^{-1}}+\frac1{1+z+x^{-1}}=1$ if $xyz=1$ Why can't I run two mysqli queries? The second one fails Embed data in a C++ program Is "princessship" a real word? Are there any other words which have the same letter 3 times consecutively? KeyListener not working