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6. Let X have a standard exponential distribution, so that $f(x) = e^{-x}$ for $x > 0$ (and zero otherwise). (a) Show that the PDF of the random variable $Y = \alpha X^{1/\beta}$, where $\alpha, \beta > 0$, is of the form $f(x) = \begin{cases} \frac{\beta}{\alpha} (\frac{x}{\alpha})^{\beta - 1} e^{-(x/\alpha)^\beta} & \text{for } x > 0, \\ 0 & \text{otherwise.} \end{cases}$ (1) This is Weibull distribution with parameters $\alpha > 0$ and $\beta > 0$. (b) Derive the quantile function $Q(u)$ of the standard exponential variable X from Part (a). (c) Let U be a standard uniform random variable and let $Q(u)$ be the quantile function derived in Part (b). Derive the CDF and the PDF of the transformed variable $Y = Q(U)$, and verify that that this variable has standard exponential distribution. [This illustrates a general phenomenon: If X is continuous with CDF $F(x)$ then we have $X \stackrel{d}{=} F^{-1}(U)$ where U is standard uniform.] (d) Show that the variable $F(X)$ is standard uniform, where X has standard exponential distribution and $F(x)$ is the CDF of X. [This illustrates a general phenomenon: If X is continuous with CDF $F(x)$ then the random variable $F(X)$ has standard uniform distribution.]

          6. Let X have a standard exponential distribution, so that $f(x) = e^{-x}$ for $x > 0$ (and zero otherwise).
(a) Show that the PDF of the random variable $Y = \alpha X^{1/\beta}$, where $\alpha, \beta > 0$, is of the form
$f(x) = \begin{cases} \frac{\beta}{\alpha} (\frac{x}{\alpha})^{\beta - 1} e^{-(x/\alpha)^\beta} & \text{for } x > 0, \\ 0 & \text{otherwise.} \end{cases}$ (1)
This is Weibull distribution with parameters $\alpha > 0$ and $\beta > 0$.
(b) Derive the quantile function $Q(u)$ of the standard exponential variable X from Part (a).
(c) Let U be a standard uniform random variable and let $Q(u)$ be the quantile function derived in Part (b). Derive the CDF and the PDF of the transformed variable $Y = Q(U)$, and verify that that this variable has standard exponential distribution. [This illustrates a general phenomenon: If X is continuous with CDF $F(x)$ then we have $X \stackrel{d}{=} F^{-1}(U)$ where U is standard uniform.]
(d) Show that the variable $F(X)$ is standard uniform, where X has standard exponential distribution and $F(x)$ is the CDF of X.
[This illustrates a general phenomenon: If X is continuous with CDF $F(x)$ then the random variable $F(X)$ has standard uniform distribution.]

6. Let X have a standard exponential distribution, so that f(x) = e^-x for x > 0 (and zero otherwise).
(a) Show that the PDF of the random variable Y = α X^1/β, where α, β > 0, is of the form
f(x) = (β)/(α) ((x)/(α))^β - 1 e^-(x/α)^β for x > 0,
0 otherwise. (1)
This is Weibull distribution with parameters α > 0 and β > 0.
(b) Derive the quantile function Q(u) of the standard exponential variable X from Part (a).
(c) Let U be a standard uniform random variable and let Q(u) be the quantile function derived in Part (b). Derive the CDF and the PDF of the transformed variable Y = Q(U), and verify that that this variable has standard exponential distribution. [This illustrates a general phenomenon: If X is continuous with CDF F(x) then we have X d= F^-1(U) where U is standard uniform.]
(d) Show that the variable F(X) is standard uniform, where X has standard exponential distribution and F(x) is the CDF of X.
[This illustrates a general phenomenon: If X is continuous with CDF F(x) then the random variable F(X) has standard uniform distribution.]

Added by Asunci-N P.

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