# Variance

• Aug 1st 2008, 12:43 AM
kenshinofkin
Variance
A dealer's profit in units of $5,000 on a new automobile is a random variable X having density function f(x,y) = 2(1-x) for -1 <= x <= 1 a) Find the vaiance in the dealer's profit. b) Demonstrate that chebyshev's inequality holds for k = 2 with the density function above. c) What is the probability that the profit exceeds$500?

a) E(2(1-x)) = -4
$\int-1,1 (-2x-1)f(x)dx$ What is f(x) in this problem? (Headbang)
b) How can I show that it holds true?
c) ???
• Aug 1st 2008, 09:40 AM
CaptainBlack
Quote:

Originally Posted by kenshinofkin
A dealer's profit in units of $5,000 on a new automobile is a random variable X having density function f(x,y) = 2(1-x) for -1 <= x <= 1 a) Find the vaiance in the dealer's profit. b) Demonstrate that chebyshev's inequality holds for k = 2 with the density function above. c) What is the probability that the profit exceeds$500?

a) E(2(1-x)) = -4
$\int-1,1 (-2x-1)f(x)dx$ What is f(x) in this problem? (Headbang)
b) How can I show that it holds true?
c) ???

$f(x)=2(1-x)\ x \in [-1,1]$

is not a density as it integrates up to 4 not 1, are you sure that the range is not $[0,1]$. Also it seems unlikely that a negative profit would be acceptable on a new car (even less a mean profit that is negative).

RonL
• Aug 1st 2008, 10:03 AM
kenshinofkin
It could be a typo. So if I go with (0,1) am I doing it right so far. f(x) = x?
So what about the other parts of the question.
b) How can I show that it holds true?
Do I do (1,2) which would show a negitive profit?
c) ???
• Aug 1st 2008, 04:57 PM
mr fantastic
Quote:

Originally Posted by kenshinofkin
A dealer's profit in units of $5,000 on a new automobile is a random variable X having density function f(x,y) = 2(1-x) for 0 <= x <= 1 Mr F says: I don't know why you have f(x , y) ....? a) Find the vaiance in the dealer's profit. b) Demonstrate that chebyshev's inequality holds for k = 2 with the density function above. [snip] a) E(2(1-x)) = -4 $\int-1,1 (-2x-1)f(x)dx$ What is f(x) in this problem? (Headbang) b) How can I show that it holds true? c) ??? b) Chebyshev's Inequality: $\Pr(|X - \mu| \geq k \sigma) \leq \frac{1}{k^2}$. So you first need to calculate: 1. $\mu = \int x f(x) \, dx = \int_{0}^{1} x [2(1 - x)] \, dx = 2 \int_{0}^{1} x - x^2 \, dx = \frac{1}{3}$. 2. $\sigma^2 = E(X^2) - \mu^2$, where $E(X^2) = \int x^2 f(x) \, dx = \int_{0}^{1} x^2 [2(1 - x)] \, dx = 2 \int_{0}^{1} x^2 - x^3 \, dx = \, .......$ (which will answer part a) by the way). Therefore $\sigma = \frac{1}{3 \sqrt{2}}$. Substitute $\mu = \frac{1}{3}$, $\sigma = \frac{1}{3 \sqrt{2}}$ and k = 2 into the left hand side of Chebyshev's Inequality: $\Pr \left ( \left |X - \frac{1}{3} \right| \geq \frac{\sqrt{2}}{3} \right)$ $= \Pr \left( X \geq \frac{\sqrt{2} + 1}{3} \right)$ $= \int_{(\sqrt{2} + 1)/3}^{1} 2 (1 - x) \, dx = \, .... \leq \frac{1}{4}$ which demonstrates the inequality. Most of this is basic application of routine definitions and formulae. NB: I reserve the right for the arithmetic details to contain errors so make sure you carefully check all numerical results given above. • Aug 1st 2008, 05:05 PM mr fantastic Quote: Originally Posted by kenshinofkin A dealer's profit in units of$5,000 on a new automobile is a random variable X having density function

f(x,y) = 2(1-x) for 0 <= x <= 1
[snip]
c) What is the probability that the profit exceeds $500? [snip] Note that a profit of$500 is equivalent to $X = \frac{1}{10}$ since X is in units of five thousand dollars.

After this it's all down hill:

$\Pr \left( X > \frac{1}{10} \right) = \int_{1/10}^{1} 2 (1 - x) \, dx = \, ....$