I am a bit confused by using Probability Mass Function and Probability Density Function.
I understand that for discrete case like Bernoulli or Binomial, we call it pmf. For continuous case like normal distribution we call it pdf.
I have encounter question involving multivariate distribution for many times and I've solved them but I just want to verify my concept.
For 2 discrete random variable $X_1$ and $X_2$ with $P$($X_1$=$x_1$,$X_2$ = $x_2$) = $f(x_1,x_2)$
if we want to find $Y$ = a$X_1$ + b$X_2$, I used to do
$P(y)$ = ∑ $P$($X_1$=$x_1$,$X_2$ = ($y$-a$x_1)/b$) = ∑$f(x_1$,($y$-a$x_1$)/$b$))
for which I know P(y) is the PMF of Y
For the continuous case if we have similar problem:
For 2 continuous random variable $X_1$ and $X_2$ with PDF = $f(x_1,x_2)$
now I want to find the distribution of $Y$ = a$X_1$ + b$X_2$
From my understanding, I just need to replace the summation with integration(Please correct me if I am wrong)
So I would obtain the PDF of y as
$f_y$($y$) = ∫ $f(x_1$,($y$-a$x_1$)/$b$)) $dx_1$
Would appreciate if anyone could tell me if I am wrong and where could I be wrong. Thanks!