Drawing Probability Density Function

Question

Can someone help me to draw this pdf? I really don't have an idea how to convert a function to pdf. Thank you

p(x | c) = 1/3 for 1 <= x <= 4 and P(c) = 0.5

Answer 1

I assume that $P(C_1\cap C_2)=0$?

Then $P(C_1)+P(C_2)=1$ and $\{C_1,C_2\}$ partitions the probability space.

You want to find $p(X)$ from the conditional probabilities.

$p(X)=P(C_1)p(X|C_1)+P(C_2)p(X|C_2)=\frac{1}{6}\mathbf{1}_{x \in [1,4]}(x)+ \frac{1}{2}\mathbf{1}_{x \in [2,3]}(x)$ This simplifies to:

$p(x)=\frac{1}{6}\mathbf{1}_{x \in \{[1,2)\cup (3,4]\}}(x)+\frac{2}{3}\mathbf{1}_{x \in [2,3]}(x)$

Its just a symmetric stepwise uniform density with tails at 1/6 and a center at 2/3.

BTW: This is called a mixture distribution. Where $P(C_1),P(C_2)$ are the mixture probabilities.