My Math Forum Questions On My Exam, did i answer correctly?

 Calculus Calculus Math Forum

 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?

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