Due to a math test I could not go to the extra class on Friday to get help on my tutorial, so I thought I'd ask here... (Itwasntme)

Quote:

If X is a geometric random variable with p = 0,5 ; for what value of k is $\displaystyle P(X \leq k) \approx 0,99$

My solution:

$\displaystyle \sum_{k=1}^{n} \left( (1-p)^{k-1} (p) \right) \approx 0,99$

$\displaystyle \sum_{k=1}^{n} \left( (1-0,5)^{k-1} (0,5) \right) \approx 0,99$

$\displaystyle \sum_{k=1}^{n} \left( (0,5)^{k-1+1} \right) \approx 0,99$

$\displaystyle \sum_{k=1}^{n} \left( (0,5)^{k} \right) \approx 0,99$

Okay now I got $\displaystyle n = 7$. But to me it is a bit of a elementary approach to just keep changing the value for $\displaystyle n$ and seeing what I get. Is there a method to solve this?

Quote:

Three identical fair coins are thrown simultaneously until all three show the same face. What is the probability that they are thrown more than three times?

My solution:

Okay now I used negative binomial distribution on this. The thing is we want 3 successes, so $\displaystyle r = 3$. I was struggling with thinking of a value for $\displaystyle k$, but logic tells me it must be a multiple of 3?

$\displaystyle P(X > 3) = 1 - P(X \leq 3)$

$\displaystyle 1 - {3-1 \choose 3-1} (0,5)^3 (1-0,5)^{3-3} $ $\displaystyle - {6-1 \choose 3-1} (0,5)^3 (1-0,5)^{6-3} - {9-1 \choose 3-1} (0,5)^3 (1-0,5)^{9-3} $

Quote:

Suppose the lifetime of an electronic component follows an exponential distribution with $\displaystyle \lambda = 0,1$

a) Find the probability that the lifetime is less than 10

b) Find the probability that the lifetime is between 5 and 15

c) Find $\displaystyle t$ such that the probability that that the lifetime is greater than $\displaystyle t$ is 0,1

My solution:

a) $\displaystyle \sum_{t=1}^{9} e^{(-0,1)(t)}$

b) $\displaystyle e^{(-0,1)(15)} - e^{(-0,1)(5)}$

c) $\displaystyle 1 - \sum_{t=1}^{n} e^{(-0,1)(t)} = 0,01$

$\displaystyle \sum_{t=1}^{n} e^{(-0,1)(t)} = 0,99$

Okay something isn't right with my number $\displaystyle c$. If i let $\displaystyle n = 1$ I get the probability is $\displaystyle 0,90$ but if I let $\displaystyle n = 2$ it ends up greater than one.

Quote:

If $\displaystyle U$ is uniform on [0;1] find the density function of $\displaystyle \sqrt{U}$

My solution:

Let $\displaystyle U$ be a uniform random variable on [0;1], and let $\displaystyle V = \sqrt{U}$

$\displaystyle F_{V}(v) = P(V \leq v)$

$\displaystyle = P(\sqrt{U} \leq v)$

$\displaystyle = P(U \leq v^2)$

$\displaystyle = f_{U} (v^2)$

$\displaystyle = v^2$

Differentiate.

$\displaystyle f_{V} (v) = 2v$

Just when I thought i was done with physics forever... (Doh)Quote:

A particle of mass $\displaystyle M$ has a random velocity, $\displaystyle V$, which is normally distributed with parameters $\displaystyle \mu = 0$ and $\displaystyle \sigma$. Find the density function of the kinetic energy.

My solution:

$\displaystyle E = \frac{1}{2} m V^2$

$\displaystyle F_{E} (e) = P(E \leq e)$

$\displaystyle = P \left( \frac{1}{2}mV^2 \leq e \right)$

$\displaystyle = P \left( V \leq \sqrt{\frac{2e}{m}} \right)$

$\displaystyle = f_{V} \left( \sqrt{\frac{2e}{m}} \right) $

$\displaystyle = \sqrt{\frac{2e}{m}}$

Differentiate. (Assuming the mass is a constant)

$\displaystyle \frac{d}{de} = \frac{1}{2 \sqrt{\frac{2e}{m}}}$

What the freak are these people on about?? (Headbang)Quote:

If the radius of a circle is an exponential random variable, find the density function of the area

My solution:

$\displaystyle A = \pi r ^2$

$\displaystyle F_{A} (a) = P(A \leq a)$

$\displaystyle = P \left( \pi r ^2 \leq a \right)$

$\displaystyle = P \left( r \leq \sqrt{\frac{a}{\pi}} \right)$

$\displaystyle = f_{r} \left( \sqrt{\frac{a}{\pi}} \right) $

$\displaystyle = \sqrt{\frac{a}{\pi}}$

Differentiate.

$\displaystyle \frac{d}{da} = \frac{1}{2 \sqrt{\frac{a}{\pi}}}$

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Thanks in advance (Nod)