PDF of a continuous random variable

[ThiagoG]

Homework Statement

Let X denote a continuous random variable with probability density function f(x) = kx³/15 for 1≤X≤2. Determine the value of the constant k.

Homework Equations

I'm not sure if this is right but I think ∫kx³/15 dx=1 with the parameters being between 2 and 1,

The Attempt at a Solution

So I did what I i showed in the relevant equations section. I got k = 4. When I plug 2 into the equation for X, my probability is greater than 1 so I know this isn't right. I'm not sure what else to do.

 

[Orodruin]

When I plug 2 into the equation for X, my probability is greater than 1 so I know this isn't right.

By this do you mean the PDF f(x) > 1? There is nothing wrong with this as f(x) is the probability density and not a probability in itself.

 

[Ray Vickson]

Homework Statement

Let X denote a continuous random variable with probability density function f(x) = kx³/15 for 1≤X≤2. Determine the value of the constant k.

Homework Equations

I'm not sure if this is right but I think ∫kx³/15 dx=1 with the parameters being between 2 and 1,

The Attempt at a Solution

So I did what I i showed in the relevant equations section. I got k = 4. When I plug 2 into the equation for X, my probability is greater than 1 so I know this isn't right. I'm not sure what else to do.

As Orodruin has indicated, f(x) can be greater than 1; the probability that $X$ lies in the interval $(x,x + \Delta x)$ is (for small $\Delta x > 0$) given by $P(x,x+\Delta x) = f(x) \cdot \Delta x + o(\Delta x)$. Here, the notation $o(h)$ means terms of higher order in small $h> 0$; that is $o(h)/h \to 0$ as $h \to 0$. In other words, for small $\Delta x > 0$ the probability is nearly proportional to $\Delta x$, with coefficient $f(x)$. Even if we have, say $f(x) = 10,$ the probability would be $10 \times 0.0001 = 1/1000 << 1$ if $\Delta x = 0.001$.