# Continuous Probability Distributions

• Mar 28th 2010, 12:32 PM
mintsharpie
Continuous Probability Distributions
For a certain lake, baseline measurements of acidity are made so that any changes caused by acid rain can be noted. The pH for water samples from the lake is a random variable X with probability density function:

http://i44.tinypic.com/30rod55.jpg

a) Sketch the curve of f(x)
b) Find the distribution function F(x) for X
c) Find the probability that the pH for a water sample from this lake will be less than 6
d) Find the probability that the pH of a water sample from this lake will be less than 5.5 given that it is known to be less than 6

Can anyone help me with this? My textbook doesn't give any examples like this, and I have no idea what to do.
• Mar 28th 2010, 12:45 PM
mr fantastic
Quote:

Originally Posted by mintsharpie
For a certain lake, baseline measurements of acidity are made so that any changes caused by acid rain can be noted. The pH for water samples from the lake is a random variable X with probability density function:

http://i44.tinypic.com/30rod55.jpg

a) Sketch the curve of f(x)
b) Find the distribution function F(x) for X
c) Find the probability that the pH for a water sample from this lake will be less than 6
d) Find the probability that the pH of a water sample from this lake will be less than 5.5 given that it is known to be less than 6

Can anyone help me with this? My textbook doesn't give any examples like this, and I have no idea what to do.

a) You should know how to draw a parabola.

b) F(x) = 0 for x < 5. $F(x) = \int_5^x \frac{3}{8} (7 - u)^2 \, du$ for $5 \leq x \leq 7$. F(x) = 1 for x > 7.

c) F(6).

d) Start by using Bayes Theorem and use your answer to b) to get the required probabilities.

All these questions will be exemplified in your textbook and/or classnotes ....
• Mar 28th 2010, 01:06 PM
mintsharpie
Quote:

Originally Posted by mr fantastic
a) You should know how to draw a parabola.

b) F(x) = 0 for x < 5. $F(x) = \int_5^x \frac{3}{8} (7 - u)^2 \, du$ for $5 \leq x \leq 7$. F(x) = 1 for x > 7.

c) F(6).

d) Start by using Bayes Theorem and use your answer to b) to get the required probabilities.

All these questions will be exemplified in your textbook and/or classnotes ....

How did you get the solution for b?

And no, they're not... all my textbook does is define what a distribution function is, F(b) = P(X ≤ b), and goes through one example that is nothing like the question I posted here. It involves finding probabilities for battery lives, using these weirdo intersections and other formulas I don't understand. I've never had such an unhelpful textbook before.
• Mar 28th 2010, 01:09 PM
mr fantastic
Quote:

Originally Posted by mintsharpie
How did you get the solution for b?

And no, they're not... all my textbook does is define what a distribution function is, F(b) = P(X ≤ b), and goes through one example that is nothing like the question I posted here. It involves finding probabilities for battery lives, using these weirdo intersections and other formulas I don't understand. I've never had such an unhelpful textbook before.

If your textbook and class notes are so unhelpful you will need to start sourcing references that are more useful to you. Using google is an excellent start. Read this: Cumulative distribution function