Question
Show that the moment-generating function of the random variable $X$ having a chi-squared distribution with $v$ degrees of freedom is$$M_{X}(t)=(1-2 t)^{-v / 2}$$
Step 1
For a chi-squared random variable with $v$ degrees of freedom, the probability density function (pdf) is given by $f(x) = \frac{1}{2^{v/2}\Gamma(v/2)}x^{v/2-1}e^{-x/2}$ for $x > 0$. Show more…
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EXAMPLES & DISCUSSION Show that the moment-generating function of the random variable X having a chi-squared distribution with v degrees of freedom is Mx(t) = (1 - 2t)^(-v/2).
The chi-squared random variable with $k$ degrees of freedom has moment-generating function $M_{X}(t)=(1-2 t)-k / 2$. Suppose that $X_{1}$ and $X_{2}$ are independent chi-squared random variables with $k_{1}$ and $k_{2}$ degrees of freedom, respectively. What is the distribution of $Y=X_{1}+X_{2} ?$
Joint Probability Distributions
Moment-Generating Functions
Prove that if a random variable X follows a standard normal distribution (with mean 0 and standard deviation 1), then Y = X^2 follows a chi-square distribution with degrees of freedom. In particular, show that M_Y(t) = M_X(t^2) = E[e^(tX^2)], which equals the moment generating function of the chi-square distribution with degrees of freedom.
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