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" Some random variables are mixtures of a discrete random variable and a continuous random variable. They have CDF of the form $$F_X=pF_{X_1}+(1-p)F_{X_2}$$ where $0\lt p \lt 1$ and $F_{X_1}$ is the CDF of a discrete random variable

and $ F_{X_{2}}$ is the CDF of a continuous random variable.

I've read that p (and 1-p) are mixture weights but what exactly are these mixture weights?

It looks like p is the probability that X is discrete and 1-p is the probability that X is continuous? Is this interpretation correct?

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Flip a weighted coin with probability $p$ of heads. If the coin comes up heads, draw $X$ from distribution $1$, if it comes up tails, draw it from distribution $2$. This describes how you would generate a variable from this distribution if you can generate variables from the two parent distributions.

Note it’s immaterial to the concept of a mixture that one of these distributions is continuous and one is discrete. You can mix any two distributions. Also, doesn’t really make sense to say “the probability the variable is discrete.” It is the probabilty it gets drawn from the discrete parent distribution.

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