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August 29th, 2014, 12:12 PM   #1
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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.
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August 29th, 2014, 12:33 PM   #2
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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.
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August 29th, 2014, 01:05 PM   #3
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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
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August 29th, 2014, 01:22 PM   #4
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Quote:
Originally Posted by Ku5htr1m View Post
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
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August 29th, 2014, 01:31 PM   #5
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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.
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August 30th, 2014, 02:07 PM   #6
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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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