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Answer by Michael Hardy for random variable follows

$$\Pr(X\le x) = 0.6 \Phi(x) + 0.4 \Phi\left( \frac x {\sqrt 2} \right)$$$$\frac d {dx} \Pr(X\le x) = 0.6 \varphi(x) + 0.4 \varphi\left( \frac x {\sqrt 2} \right) \cdot \frac 1 {\sqrt 2}$$(The chain...

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Answer by Augustin for random variable follows

Let $Y$ a Bernoulli distributed random variable with parameter $p=0.6$, that is $P(Y=1)=1-P(Y=0)=0.6$.Then you have the conditional distribution of $X_1$ given $Y=1$: $$X_1\mid...

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random variable follows

I have a random variable $x_1$ that follows the normal distribution with mean 0 and variance 1 with probability 0.6 and follows the normal distribution with mean 0 and variance 2 with probability 0.4....

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