I am struggling on the following problem (altered for better understanding): Let's say the height of a father $x_F$ is drawn from a normal distribution $X_{F} \sim \mathcal{N}(\mu,\sigma_F^2)$ and the height of the son $x_s$ is drawn from $X_{S} \sim \mathcal{N}(x_F,\sigma_S^2)$, what is now the probability that both are shorter than a specific value $h$?

So basically $P(X_s<h \cap X_F<h)$

My attempt: $P(X_s<h \cap X_F<h) = P(X_s<h|X_F<h)*P(X_F<h)$

For $P(X_s<h|X_F=x_F)=\int_{-\infty}^{h} \frac{1}{\sqrt{2\pi\sigma_S^2} } exp( -\frac{(x-x_F)^2}{2\sigma_S^2})dx$

But what about the continuous case? What would change if we start at grandpa?


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