Question

What is the probability density function of a chi-squared distribution?

Answer 1

Hint: Let us first understand about the chi-square distribution. The chi-square distribution also called Chi-squared distribution (${\chi ^2}$ -distribution) with $n$ degrees of freedom is the distribution of a sum of the squares of k standard normal variables. It is a special case of the gamma distribution and is used in statistical hypothesis testing.
The probability density function (also called probability function) is defined for continuous random variables lying between a certain range of values. That is, the probability density function for continuous random variables that takes value between certain limits say $a$ and $b$ is calculated by the formula,
\[P(a < x < b)\] or $P(x) = \int\limits_a^b {f(x)dx} $
The probability density function is non-negative for all $x$ $(f(x) \geqslant 0)$ .
Also, it is noted that $\int\limits_{ - \infty }^\infty {f(x)dx = 1} $.
Formula Used:
The probability density function for chi-square distribution with $n$ degrees of freedom is as follows.
\[P(x) = \dfrac{{{x^{\dfrac{n}{2} - 1}}{e^{\dfrac{{ - x}}{2}}}}}{{\Gamma (\dfrac{1}{2}n){2^{\dfrac{n}{2}}}}}\] for all $x \in [0,\infty )$,
Where, $\Gamma \left( x \right)$ is a gamma function .

Complete step-by-step solution:
A chi-square random variable (denoted symbolically \[{\chi ^2}_n\] ) with $n$ degrees of freedom is a continuous random variable for all possible values in $[0,\infty )$ .
The chi-square distribution has the probability density function (PDF) given by
$f(x)\left\{ \begin{gathered}
\dfrac{{{x^{\dfrac{n}{2} - 1}}{e^{\dfrac{{ - x}}{2}}}}}{{\Gamma (\dfrac{n}{2}){2^{\dfrac{n}{2}}}}},if(x \geqslant 0) \\
  0,if(x \leqslant 0) \\
\end{gathered} \right\}$
Here,$\Gamma \left( x \right)$ is a gamma function,
Where, $\Gamma \left( x \right)$$ = \int\limits_0^\infty {{e^{ - t}}} {t^{x - 1}}dt$.

Note: We shall go through some properties of the chi-square distribution which are as follows:
1). The mean of the distribution and the number of degrees of freedom are equal (i.e.)$\mu = n$ .
2). The variance of the distribution is equal to the two times the number of degrees of freedom (i.e.)${\sigma ^2} = 2 \times n$ .
3). If the degrees of freedom increase, the Chi-square curve approaches a normal distribution.