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 August 29th, 2014, 12:12 PM #1 Member   Joined: Sep 2013 Posts: 83 Thanks: 0 Questions On My Exam, did i answer correctly? The density of a certain random variable is defined by $\displaystyle f(x)=\left\{\begin{matrix} C\sqrt{x} & if \, 0\leq x\leq 1 \\ 0 & if\, otherwise \end{matrix}\right.$ where C is a constants. What is the value of C? (a) 1 -----my answer (b) 1/3 (c) 2/3 (d) 3/2 Which of the following functions could qualify as the density function of a random variable? (a) ----- my answer $\displaystyle f(x)=\left\{\begin{matrix} 0.5e^{-0.5x} & if \: x>0 \\ 0 & if\, otherwise \end{matrix}\right.$ (b) $\displaystyle f(x)=e^{x}$ (c) $\displaystyle f(x)=\frac{1}{x}$ (d) $\displaystyle f(x)=\frac{1}{x^{2}}$ The value of the definite integral $\displaystyle \int_{0}^{x}(t^{2}x+1)dt$ (a) depends on t and x ---- my answer (b) depends only on t but not on x (c) depends only on x but not on t (d) doesn't depend on either x or t Last edited by Ku5htr1m; August 29th, 2014 at 12:25 PM. August 29th, 2014, 12:33 PM #2 Math Team   Joined: Dec 2013 From: Colombia Posts: 7,674 Thanks: 2654 Math Focus: Mainly analysis and algebra 1) Incorrect. d) is the correct answer. $$\int_0^1 Cx^\tfrac12 \,\mathbb{d}x = \tfrac23 Cx^\tfrac32 \bigg|_0^1 = \tfrac23 C$$ We require the integral to be equal to 1, so $C = \tfrac32$. 2) There are no intervals with most of them! But your answer is correct. b blows up as $x \to \infty$ while the other two blow up at $x = 0$. 3) Incorrect. The variable $t$ is a dummy variable for the integral. The solution will still have the variable $x$, not the variable $t$. So c) is the correct answer. Last edited by v8archie; August 29th, 2014 at 12:37 PM. August 29th, 2014, 01:05 PM #3 Member   Joined: Sep 2013 Posts: 83 Thanks: 0 Ok thank you for the answer and the explanation ! Let me continue with a couple of exam question I was not sure on. The vectors $\displaystyle \begin{pmatrix} 1\\ 5\\ 3\\ \end{pmatrix} \begin{pmatrix} 1\\ 2b\\ 2\\ \end{pmatrix}$ (a) may be linearly dependent or linearly independent, depending on the value of b (b) are linearly dependent ---- my answer (c) are linearly independent (d) none of the preceding You are joining a game that involves flipping a fair coin three times. If the coin comes up heads two or three times, you win €500; otherwise, you lose €400. What are you expected earnings in the game? (a) €0 (b) € 50 ---- my answer (c) € 100 (d) € -50 In a certain binomial tree model for a stock price, the stock price today is €100.Each day from now on, independently of each other day, the price goes up or down €1, up with probability 0.7 and down with probability 0.3. What is the expected stock price three days from today? (a) €96 (b) €98 (c) €100.6 ---my answer (d) none of the preceding Let $\displaystyle k\epsilon \mathbb{R}, k\neq 0. Then, the\: matrix : A= \begin{pmatrix} 1/k & 0 & 1 \\ -1 & 0 & 4\\ 0& 1 & k \end{pmatrix}$ has rank: (a) 3 for all k ---- my answer (b) 3 if $\displaystyle \neq$ -4 and 2 if k=-4 (c) 3 if $\displaystyle \neq$ -4, 2 if k=-4 and 1 if k =1 (d) none of the preceding August 29th, 2014, 01:22 PM   #4
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 Originally Posted by Ku5htr1m Ok thank you for the answer and the explanation ! Let me continue with a couple of exam question I was not sure on. The vectors $\displaystyle \begin{pmatrix} 1\\ 5\\ 3\\ \end{pmatrix} \begin{pmatrix} 1\\ 2b\\ 2\\ \end{pmatrix}$ (a) may be linearly dependent or linearly independent, depending on the value of b (b) are linearly dependent ---- my answer (c) are linearly independent (d) none of the preceding You are joining a game that involves flipping a fair coin three times. If the coin comes up heads two or three times, you win €500; otherwise, you lose €400. What are you expected earnings in the game? (a) €0 (b) € 50 ---- my answer (c) € 100 (d) € -50 In a certain binomial tree model for a stock price, the stock price today is €100.Each day from now on, independently of each other day, the price goes up or down €1, up with probability 0.7 and down with probability 0.3. What is the expected stock price three days from today? (a) €96 (b) €98 (c) €100.6 ---my answer (d) none of the preceding Let $\displaystyle k\epsilon \mathbb{R}, k\neq 0. Then, the\: matrix : A= \begin{pmatrix} 1/k & 0 & 1 \\ -1 & 0 & 4\\ 0& 1 & k \end{pmatrix}$ has rank: (a) 3 for all k ---- my answer (b) 3 if $\displaystyle \neq$ -4 and 2 if k=-4 (c) 3 if $\displaystyle \neq$ -4, 2 if k=-4 and 1 if k =1 (d) none of the preceding
You got 1 out of 4 correct. Which is the expected average if you were just guessing. The answers are c b d b August 29th, 2014, 01:31 PM #5 Member   Joined: Sep 2013 Posts: 83 Thanks: 0 yes I was more or less guessing on a lot of the example I mentioned here... ..could you give some explanation to how you arrived at the right answers? Last edited by Ku5htr1m; August 29th, 2014 at 01:49 PM. August 30th, 2014, 02:07 PM #6 Member   Joined: Sep 2013 Posts: 83 Thanks: 0 Finding the local maximum & minimum ? $\displaystyle f(x,y) = (x+y)^2+(y-1)^2$ The point x=-1 and y = 1 Is that a local maximum or a local minimum ? $\displaystyle f(x_{1},x_{2}) = 3x_{1}-3x_{2}-x^3+x^3$ The point x1 =1 and x2= -1 is a local maximum or local minimum? Tags answer, correctly, exam, questions Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post Bucephalus Algebra 2 January 26th, 2012 11:08 PM chemicalenginner Calculus 3 January 7th, 2011 06:44 PM xdeathcorex Algebra 5 November 1st, 2010 05:15 PM ahhaaa Probability and Statistics 1 May 18th, 2010 04:23 PM cheatzero Algebra 4 July 7th, 2009 07:00 AM

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