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Find the pdf of $Y=\frac{X}{X+1}$

The first step is to think about the support of $Y$. If $X \in [0,1]$, then what is the range of $Y = f(X) = X/(X+1)$? When $X = 0$, we have $Y = 0$. But when $X = 1$, then $Y = 1/2$. Intuition ...
heropup's user avatar
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2 votes
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PDF of $Y=g(X)$ when $X\sim N(0,1)$. $g(X)$ is a piecewise function where each part is constant.

Your expression for the CDF of $Y$ is close, but should be $$F_{Y}\left(y\right)=\begin{cases} 0, & y<-1\\ 1/2, & -1\leq y<1\\ 1, & y\geq1 \end{cases}.$$ Since $Y$ is discrete, it ...
AOS's user avatar
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$\mathbb{E}[X^2]\leq k \mathbb{E}[X]^2$, upper bound second moment from first moment

Despite your "It's easy to show ...", Bernoulli random variables with $\mathbb P(X=1)=p$ are a counter-example: $$\dfrac{\mathbb{E}[X^2]}{ \mathbb{E}[X]^2} = \dfrac{p}{p^2} = \dfrac1p$$ and ...
Henry's user avatar
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2 votes
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How to Determine Independence of Events Using Probability

Independent has a very specific meaning in probability, namely events $A$ and $B$ are independent if and only if $Pr(A|B)=Pr(A)$. A consequence of this, which comes from the rule $Pr(A|B)=\frac{Pr(A\...
Red Five's user avatar
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1 vote

MIT Statistic For Applications course Question 1

It's Markov's inequality. thanks Matthew Towers.
tom tom's user avatar
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1 vote
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Expected value and average value problem

If $$1-\Pr\left\{ 0.64\leq X\leq0.66\right\}$$ $$=1-\Pr\left\{ \dfrac{0.64-\mu}{\sigma/\sqrt{n}}\leq\dfrac{X-\mu}{\sigma/\sqrt{n}}\leq\dfrac{0.70-\mu}{\sigma/\sqrt{n}}\right\},$$ then $$\Pr\left\{ 0....
AOS's user avatar
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1 vote

Suppose $Y\backsim N(0, 1)$ and $X\backsim N(0, Y^{-2})$. Show that $X$ has the standard Cauchy distribution.

If $$X \mid Y \sim \operatorname{Normal}(0, Y^{-2}),$$ then the variance of $X \mid Y$ is $1/Y^2 > 0$, and the conditional density of $X \mid Y$ is $$f_{X \mid Y}(s \mid t) = \frac{|t|}{\sqrt{2\pi}}...
heropup's user avatar
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What to call a sequence of Bernoulli trials with different probabilities?

It is Poisson Binomial Distribution : In Probability theory and Statistics, the Poisson Binomial Distribution is the Discrete Probability Distribution of a sum of independent Bernoulli trials that are ...
Prem's user avatar
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1 vote

What to call a sequence of Bernoulli trials with different probabilities?

If $p$ is fixed and all the experiments are Bernoulli trials then we can call it a Binomial Distribution. If $p$ is not fixed, then it is more useful to consider the Bernoulli trials as separate ...
Red Five's user avatar
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