Intereting Posts

Isomorphism between tensor product and quotient module.
Finding the Circulation of a Curve in a Solid. (Vector Calculus)
Norm induced by convex, open, symmetric, bounded set in $\Bbb R^n$.
gcd and order of elements of group
Why is Hodge more difficult than Tate?
Showing non-cyclic group with $p^2$ elements is Abelian
Triangular matrices proof
The “Easiest” non-smoothable manifold
Find $C$, if $A=CBC$, where $A$,$B$,$C$ are symmetric matrices.
Evaluating the integral with trigonometric integrand
How can find this sequence $ a_{n+1}=a_{n}+na_{n-1},$
Approximating $\pi$ using Monte Carlo integration
Dual residual for linearized ADMM
Is there a “positive” definition for irrational numbers?
Proving The Average Value of a Function with Infinite Length

I need to prove that the age behavior of the Gamma Distribution with probability density function $$ f(x)=\frac{\lambda^{\alpha}x^{\alpha-1}e^{-\lambda x}}{\Gamma(\alpha)}$$

For $x\geq 0$, $\lambda, \alpha >0$; I.e., the conditional pprobability $P(X>x+t|X>t)$, increases in $t$ whenever $\alpha >1$ and decreases in $t$ whenever $0<\alpha <1$.

Thus far, I have the following: $$P(X>x+t|X>t)=\frac{P(X>x+t)\cap P(X>t)}{P(X>t)} = \frac{P(X>x+t)}{P(X>t)} =\frac{\int_{x+t}^{\infty}\frac{\lambda^{\alpha}u^{\alpha-1}e^{-\lambda u}}{\Gamma(\alpha)}du}{\int_{t}^{\infty}\frac{\lambda^{\alpha}u^{\alpha-1}e^{-\lambda u}}{\Gamma(\alpha)}du}.$$

I didn’t know how to go further, until my professor hinted to take the derivative of the last part with respect to $\alpha$. If it had been w.r.t. $t$, then I could use the fundamental theorem of calculus maybe to help me figure out the derivatives of those integrals.

- If $f(0) = f(1)=0$ and $|f'' | \leq 1$ on $$, then $|f'(1/2)|\le 1/4$
- Is $\sqrt{\left(\operatorname{Si}(x)-\frac\pi2\right)^2+\operatorname{Ci}(x)^2}$ monotonic?
- Finding the limit $\lim_{x\rightarrow 0^{+}}\frac{\int_{1}^{+\infty}\frac{e^{-xy}\quad-1}{y^3}dy}{\ln(1+x)}.$
- Two Strictly Convex Functions with Contact of Order 1
- Second derivative of a vector field
- How should I calculate the $n$th derivative of $f(x)=x^x$?

I do know that $\Gamma(\alpha)=\int_{0}^{\infty}y^{\alpha – 1}e^{-y}dy = (\alpha-1)\Gamma(\alpha -1)$, but how do I take the derivative of that w.r.t. $\alpha$? At least in a meaningful way that I should help me towards my ultimate goal, which is to prove that the conditional probability increases in $t$ for $\alpha >1$ and decreases in $t$ for $0<\alpha<1$.

Could somebody please help me figure out how to finish this proof? Thank you for your time and patience.

I’ve even tried even entering this all into Maple and letting it do the calculations for me; the problem with this is I keep getting error messages because I don’t know how to define $\alpha$ properly! I’m so lost and desperately need to see how these derivatives are done. Even if you don’t want to work out the whole thing for me, I’d settle just to see how to find the derivative of the integral in the numerator of my last expression. **I would accept an answer with just this part worked out in detail.** Please.

- Power of a function is analytic
- What justifies writing the chain rule as $\frac{d}{dx}=\frac{dy}{dx}\frac{d}{dy}$ when there is no function for it to operate on?
- Derive recursion formula for an integral
- Matrix derivative $(Ax-b)^T(Ax-b)$
- Prove that $\sin x \cdot \sin (2x) \cdot \sin(3x) < \tfrac{9}{16}$ for all $x$
- Vector derivation of $x^Tx$
- Prob. 15, Chap. 5 in Baby Rudin: Prove that $M_1^2\leq M_0M_2$, where $M_0$, $M_1$, and $M_2$ are the lubs, resp., of …
- How to approach solving $\int_0^{\pi/2} \ln(a^2 \cos^2 x +b^2 \sin^2 x ) dx$
- The derivative of $e^x$ using the definition of derivative as a limit and the definition $e^x = \lim_{n\to\infty}(1+x/n)^n$, without L'Hôpital's rule
- Proof: Derivative of $(-1)^{x}$

Fix some $a>0$ and consider, for every $x>0$, $$g(x)=x^{a-1}e^{-x}\qquad G(x)=\int_x^\infty g(y)dy$$ Then the question asks to study the sense of variation of the function $R_x$ defined on $t>0$ as $$R_x(t)=\frac{G(x+t)}{G(t)}$$ Equivalently, considering the logarithmic derivative of $R_x$ and using the fact that $G’=-g$, one is looking for the sign of $$h(t,x)=g(t)G(x+t)-g(t+x)G(t)$$

The change of variable $y\to zy$ in $G(z)$ yields, for every $z>0$, $$G(z)=z\int_1^\infty g(zy)dy=z^a\int_1^\infty y^{a-1}e^{-zy}dy$$ hence, applying this to $z=t$ and to $z=x+t$, one sees that $$h(t,x)=t^{a-1}e^{-t}(x+t)^a\int_1^\infty y^{a-1}e^{-(x+t)y}dy-(t+x)^{a-1}e^{-t-x}t^a\int_1^\infty y^{a-1}e^{-ty}dy$$ which has the sign of $$j(t,x)=(x+t)\int_1^\infty y^{a-1}e^{-(x+t)y}dy-e^{-x}t\int_1^\infty y^{a-1}e^{-ty}dy$$ Now, integrating by parts, for every $z>0$, $$z\int_1^\infty y^{a-1}e^{-zy}dy=e^{-z}+(a-1)\int_1^\infty y^{a-2}e^{-zy}dy$$ hence $$j(t,x)=e^{-x-t}+(a-1)\int_1^\infty y^{a-2}e^{-(x+t)y}dy-e^{-x}\left(e^{-t}+(a-1)\int_1^\infty y^{a-2}e^{-ty}dy\right)$$ which has the sign of $(a-1)k(t,x)$ with $$k(t,x)=\int_1^\infty y^{a-2}e^{-(x+t)y}dy-e^{-x}\int_1^\infty y^{a-2}e^{-ty}dy$$ This is also $$k(t,x)=\int_1^\infty y^{a-2}(e^{-xy}-e^{-x})e^{-ty}dy$$ The parenthesis in the integral is always negative hence $k(t,x)<0$ thus $R_x(t)$ is an increasing function of $t$ if $a<1$ and a decreasing function of $t$ if $a>1$.

*Edit:* To sum up the sequence of computations above,

$$\frac{\partial R_x(t)}{\partial t}=(1-a)\frac{g(t)g(x+t)}{G(t)^2}\int_0^\infty e^{-xz}\int_z^\infty(y+1)^{a-2}e^{-ty}dy\,dz$$

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- Three Variables-Inequality with $a+b+c=abc$
- Let $p$ be a prime, then does $p^{\alpha} \mid |G| \Longrightarrow p \mid Aut(G)$?
- Prove that a set $A$ is $\mu^\star$ measurable is and only if $\mu^\star (A) = l(X) – \mu^\star(A^{c})$
- Are strict local minima of a general function always countable?
- On a system of equations with $x^{k} + y^{k} + z^{k}=3$ revisited
- Show that $15\mid 21n^5 + 10n^3 + 14n$ for all integers $n$.
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- How to solve $x^3=-1$?
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- Why does spectral norm equal the largest singular value?