Quiz — Conditional Probability ENGR200

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1) For a joint discrete pmf $f_{XY}(x,y)$:

$$\sum_x\sum_y f_{XY}(x,y)=1.$$

2) The marginal pmf is obtained by summing the joint over the other variable:

$$f_X(x)=\sum_y f_{XY}(x,y).$$

3) The conditional pmf equals $f_Y(y)/f_X(x)$.

4) For a conditional pdf,

$$\int_{-\infty}^{\infty} f_{Y|X}(y|x)\,dy = 1 \quad\text{whenever } f_X(x)>0.$$

5) If $X$ and $Y$ are independent, then $p_{Y|X}(y|x)=p_Y(y)$.

6) If $p_{Y|X}(y|x)=p_Y(y)$ holds for just one $x$, that’s enough for independence.

7) Law of total probability (discrete):

$$p_Y(y)=\sum_x p_{Y|X}(y|x)p_X(x).$$

8) Bayes’ rule:

$$p_{X|Y}(x|y)=\frac{p_{Y|X}(y|x)p_X(x)}{p_Y(y)},\quad p_Y(y)>0.$$

9) In a joint pmf table, negative cells (e.g. -0.01) are acceptable if total=1.

10) For events $A,B$ with $P(B)>0$:

$$P(A|B)=\frac{P(A\cap B)}{P(B)}.$$

11) If $f_X(x_0)=0$, then $p_{Y|X}(y|x_0)=0$ for all $y$.

12) If $X,Y$ independent, then

$$F_{XY}(x,y)=F_X(x)+F_Y(y).$$

13) If $f_{XY}(x,y)=f_X(x)f_Y(y)$ for all $x,y$, then $\operatorname{Cov}(X,Y)$ can be nonzero.

14) If $\operatorname{Cov}(X,Y)=0$, then $X$ and $Y$ must be independent.

15) For continuous $X,Y$: $P(X=a,Y=b)>0$ whenever $f_{XY}(a,b)>0$.

Score: 0/15 correct