Let f(x)=k(3 x-x^2) if 0⩽x ⩽3 and f(x)=0 if x 3 . (a) For what value of k is f a probabil

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Let f(x)=k(3 x-x^2) if 0⩽x ⩽3 and f(x)=0 if x<0 or x>3 . (a)...

Let $f(x)=k\left(3 x-x^{2}\right)$ if 0$\leqslant x \leqslant 3$ and $f(x)=0$ if $x<0$ or $x>3 .$
(a) For what value of $k$ is $f$ a probability density function?
(b) For that value of $k,$ find $P(X>1)$ .
(c) Find the mean.

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Key Concepts

Probability Density Function

A probability density function (pdf) describes the likelihood of a continuous random variable taking on a given value. It is defined such that the probability of the variable falling within a certain interval can be found by integrating the pdf over that interval, and the total area under the curve across the entire domain must equal 1.

Normalization Condition

The normalization condition ensures that the integral of the pdf over its entire domain equals 1. When a function is defined with an unknown constant, this condition is used to solve for that constant so that the function properly qualifies as a pdf.

Calculating Probabilities via Integration

For continuous random variables, the probability that the variable exceeds or lies within a certain value or interval is determined by integrating the pdf over that specified range. This method is fundamental in finding probabilities such as P(X > a) or P(b < X < c).

Expected Value (Mean)

The expected value, or mean, of a continuous random variable is calculated by integrating the product of the variable and its pdf over its entire range. This value represents the theoretical average outcome or the center of mass of the probability distribution.

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