Thread: Probability of a random variable

1. Probability of a random variable

The reading given by a thermometer calivrated in ice water is a random variable with probablility density function

f(x) { k(1-x) -1

{ 0 otherwise where k is a constant

find the value of k.

What is the probablity that the temperatiure reading is greater than 0C?

What is the mean reading and the standard devation?

2. Originally Posted by jabba1
The reading given by a thermometer calivrated in ice water is a random variable with probablility density function

f(x) { k(1-x) -1

{ 0 otherwise where k is a constant

find the value of k.

What is the probablity that the temperatiure reading is greater than 0C?

What is the mean reading and the standard devation?
Since I can read your density function to find k you need to solve this integral

$\displaystyle \int_{a}^{b}f(x)dx=1$ the limits will be the where $\displaystyle f(x)$ is not zero.

the mean

$\displaystyle \mu=\int_{a}^{b}xf(x)dx$

$\displaystyle \sigma^2=\int_{a}^{b}(x-\mu)^2f(x)dx$

3. i just noticed a mistake...it should say f(x) { k(1-x) -1<x<1

4. Originally Posted by jabba1
i just noticed a mistake...it should say f(x) { k(1-x) -1<x<1
Well then use that: $\displaystyle a=-1~\&~b=1$

5. Originally Posted by jabba1
i just noticed a mistake...it should say f(x) { k(1-x) -1<x<1
okay my original post still stands you need to solve this

$\displaystyle \displaystyle \int_{-1}^{1}k(1-x)dx =1 \iff k \int_{-1}^{1}(1-x)dx=1$
$\displaystyle \displaystyle k\left[ -\frac{(1-x)^2}{2}\bigg|_{-1}^{1}\right]=1 \iff 2k=1 \iff k=\frac{1}{2}$

Now $\displaystyle f(x)=\begin{cases} \frac{1-x}{2} , \text{ if }x \in[-1,1] \\ 0, \text{ otherwise }\end{cases}$

Now use this $\displaystyle f(x)$ and the same limits of integration to compute the other 2 integrals.

6. I'm still a little confused on how exactly to use the equation. I think that's my biggest problem

7. Originally Posted by jabba1
I'm still a little confused on how exactly to use the equation. I think that's my biggest problem
$\displaystyle \displaystyle \mu=\int_{-1}^{1}x\left(\frac{1-x}{2} \right)dx=\frac{1}{2}\int_{-1}^{1}x-x^2dx$

Then use this for the 2nd one!

8. thank you! i plugged in the number and for the first one i got .5 and for the second i got .45