What is this hybrid(mixed) random variable’s variance?

X ∼ Uniform(a,b), a<b (Discrete) where f(x)=1/n where n=b-a+1 and Y ∼ Uniform(c,d), c<d (Continuous) where g(y)=1/d-c. X and Y are independent. Let z = x - y. I was able to find the E(Z), however I couldn’t derive its variance. What is the Var(z)? I couldn’t write the integral from of its variance to begin evaluating. How do I write its variance with integral and sigma? Is the following process accurate?

• If this is an exercise or homework, please add the 'self-study' tag and explain what are your difficulties with solving the question. If not please provide more context. Dec 12, 2023 at 10:01
• Although you can write an expression for the variance in terms of integrals, they are messy due to the piecewise linear nature of the PDF of $Z.$ Why not just apply the simple laws of variance, such as $\operatorname{Var}(X-Y)=\operatorname{Var}(X)+\operatorname{Var}(Y)$ when $X$ and $Y$ are independent?
– whuber
Dec 12, 2023 at 15:05
• Yeah but how would that integral look, that is what I am trying to find. Dec 12, 2023 at 15:26
• Back up, then, and start with a preliminary, simpler question. For instance, let $a=0$ and $b=1$ so that $X$ is a Bernoulli$(1/2)$ variable and let $c=0$ so that $Y/d$ has a Uniform$(0,1)$ distribution. If you have any difficulty finding the PDF of $X+Y$ (basic definitions will get you there if you first find the CDF), then show us your progress and ask for help there. Otherwise, what you learn from this simpler exercise will readily generalize to the full distribution of $X+Y$ in the general case and you can apply the definition of variance.
– whuber
Dec 12, 2023 at 15:59