# Conditional probability of $Y > 250$ given $X$ and $Y$ is either greater than $250$ or it's not.

Suppose $$Y=\beta_1 X + \beta_2 X^2$$ for some real numbers $$\beta_1$$ and $$\beta_2$$ where $$X$$ is a random variable with real values greater than $$0$$ and $$Y$$ is greater than $$0$$.

What is the probability $$P[Y>250 | X]$$ if you consider $$Y$$ in a binary way as being either $$>250$$ or $$<250$$?

(This is a simplified version of equation $$(7.3)$$ in Introduction to Statistical Learning by Witten and Hastie on page $$268$$.)

I think we have that $$P[Y>250 | X]=1-P[Y\leq 250 | X]$$.

The text seems to claim that such a probability would be $$\frac{e^{\beta_1 X + \beta_2 X^2}}{1+e^{\beta_1 X + \beta_2 X^2}}$$ but I'm not following.

• I don't get the question. $Y$ is $X$ measurable, hence $\mathbb P( \{Y> 250\} | X) := \mathbb E[ 1_{(250,+\infty)}(Y) | X] = 1_{(250,+\infty)}(Y)$. Note that conditional expectation/probability is a random variable and $\mathbb E[Z|\mathcal G] = Z$ if $Z$ is $\mathcal G$ measurable. Oct 7, 2020 at 10:55
• The textbook claims that $P[Y>250|X]=\frac{e^{\beta_1 X + \beta_2 X^2}}{1+e^{\beta_1 X + \beta_2 X^2}}$ and that they treat $Y$ as binary in that of all the values of $Y$ that are possible, they separate it as those that are greater than or less than 250. I just don't see how they got their answer. Oct 7, 2020 at 11:18
• In your notation what does $1_{(250, +\infty)}(Y)$ mean? Oct 7, 2020 at 11:19
• $1_A$ stands for indicator function, that is $1_A(x)=1$ if $x \in A$ and $1_A(x)=0$ otherwise Oct 7, 2020 at 11:20
• What resource are you using? Oct 7, 2020 at 11:22