Question

6. Define the function $h_X(x)$ to be the hazard function of the continuous random variable X. Recall our notation for the probability distribution function and cumulative distribution function of X is $f(x)$ and $F(x)$, respectively. Given our recollection, the hazard function may be expressed in terms of both $f$ and $F$ in the formula $h_X(x) = \frac{f(x)}{1 - F(x)}$ 1. Verify by computation or citing the relevant rule in calculus that $h_X(x) = -\frac{d}{dx} \log(1 - F(x))$ 2. Suppose that $X \sim Weibull(\alpha, \beta)$, compute $h_X(x)$.

          6. Define the function $h_X(x)$ to be the hazard function of the continuous random variable X. Recall our notation for the probability distribution function and cumulative distribution function of X is $f(x)$ and $F(x)$, respectively. Given our recollection, the hazard function may be expressed in terms of both $f$ and $F$ in the formula

$h_X(x) = \frac{f(x)}{1 - F(x)}$

1. Verify by computation or citing the relevant rule in calculus that $h_X(x) = -\frac{d}{dx} \log(1 - F(x))$

2. Suppose that $X \sim Weibull(\alpha, \beta)$, compute $h_X(x)$.

6. Define the function hX(x) to be the hazard function of the continuous random variable X. Recall our notation for the probability distribution function and cumulative distribution function of X is f(x) and F(x), respectively. Given our recollection, the hazard function may be expressed in terms of both f and F in the formula

hX(x) = (f(x))/(1 - F(x))

1. Verify by computation or citing the relevant rule in calculus that hX(x) = -(d)/(dx)log(1 - F(x))

2. Suppose that X ∼ Weibull(α, β), compute hX(x).

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