PDF of a continuous random variable

So the fact that ##f(2) > 1## is not relevant.In summary, we are trying to determine the value of the constant ##k## in a continuous random variable with probability density function ##f(x) = kx^3/15## for ##1 \leq X \leq 2##. After setting up the relevant equation, we find that ##k = 4## but this may not seem right since ##f(2) ## is greater than 1. However, this is not a problem as the probability density function can take values greater than 1.
  • #1
ThiagoG
15
0

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.
 
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  • #2
ThiagoG said:
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.
 
  • #3
ThiagoG said:

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##.
 

What is a PDF of a continuous random variable?

A PDF (probability density function) of a continuous random variable is a mathematical function that describes the probability distribution of that variable. It is used to calculate the likelihood of a particular outcome of a continuous random variable.

How is a PDF related to a continuous random variable?

A PDF represents the probability distribution of a continuous random variable. It shows the different possible values of the variable and their corresponding probabilities.

What is the difference between a PDF and a CDF of a continuous random variable?

A PDF represents the probability density of a continuous random variable, while a CDF (cumulative distribution function) represents the cumulative probability of the variable. In other words, the PDF shows the likelihood of a specific value occurring, while the CDF shows the likelihood of a value less than or equal to a given value occurring.

How is a PDF used in statistics?

A PDF is used to determine the likelihood of a particular outcome of a continuous random variable. It is also used to calculate other statistical measures such as mean, median, and variance.

What are the properties of a PDF of a continuous random variable?

The properties of a PDF include: it must always be non-negative, the integral over the entire range of values must equal 1, and it can never be greater than 1 for any value. Additionally, the area under the curve between any two points represents the probability of the variable taking on a value within that range.

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