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dusya [7]
3 years ago
14

Let X and Y be independent continuous random variables that are uniformly distributed on (0,1). Let H=(X+2)Y. Find the probabili

ty P(lnH≥z) where z is a given number that satisfies ez<2. Your answer should be a function of z.Hint: Condition on X.P(lnH≥z1)= ? Let X be a standard normal random variable, and let FX(x) be its CDF. Consider the random variable Z=FX(X). Find the PDF fZ(z) of Z. Note that fZ(z) takes values in (0,1) .fZ(z)= ?
Mathematics
1 answer:
belka [17]3 years ago
4 0

Looks like there are two distinct problems here.

  • "Let X and Y be independent ..."

P(\ln H\ge z)=P(\ln((X+2)Y)\ge z)=P((X+2)Y\ge e^z)=P\left(Y\ge\dfrac{e^z}{X+2}\right)

Conditioning on X involves interpreting this probability for all possible fixed values of X=x:

P\left(Y\ge\dfrac{e^z}{X+2}\right)=\displaystyle\int_xP\left(Y\ge\dfrac{e^z}{X+2}\text{ and }X=x\right)

P\left(Y\ge\dfrac{e^z}{X+2}\right)=\displaystyle\int_xP\left(Y\ge\dfrac{e^z}{X+2}\mid X=x\right)P(X=x)

We have x\in(0,1), and over this domain P(X=x)=1. Since X=x is fixed, we can omit the conditioning notation to leave us with

P\left(Y\ge\dfrac{e^z}{X+2}\right)=\displaystyle\int_0^1P\left(Y\ge\dfrac{e^z}{x+2}\right)\,\mathrm dx

Y has CDF

P(Y\le y)=F_Y(y)=\begin{cases}0&\text{for }y1\end{cases}

Then we can write our integral as

P\left(Y\ge\dfrac{e^z}{X+2}\right)=\displaystyle\int_0^1\left(1-F_Y\left(\frac{e^z}{x+2}\right)\right)\,\mathrm dx

P\left(Y\ge\dfrac{e^z}{X+2}\right)=\displaystyle\int_0^1\left(1-\frac{e^z}{x+2}\right)\,\mathrm dx

P\left(Y\ge\dfrac{e^z}{X+2}\right)=x-e^z\ln(x+2)\bigg|_{x=0}^{x=1}

P\left(Y\ge\dfrac{e^z}{X+2}\right)=\boxed{1-e^z(\ln2-\ln3)}

  • "Let X be a standard normal ..."

Using the method of distribution functions, we have

F_Z(z)=P(Z\le z)=P(F_X(X)\le z)=P(X\le {F_X}^{-1}(z))=F_X({F_X}^{-1}(z))=z

\implies\boxed{f_Z(z)=\begin{cases}1&\text{for }z\in(0,1)\\0&\text{otherwise}\end{cases}}

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