Chapter 5 – Quiz 1 – E600 Mathematics

1. Two random variables X and Y are jointly continuous. Their joint density is given by $\phi_{X,Y}(x,y)$ . Which of the following is false?

Their joint distribution is given by $\int_B \phi_{X,Y}(x,y)dx\:dy$

The marginal distribution of X is given by $\int_{\mathbb{R}}\int_B \phi_{X,Y}(x,y)dy\:dx$

If the two random variables are independent, their joint density simplifies to $\phi_{X,Y}(x,y) = \phi_X(x) \phi_Y(y)$

The cumulative distribution function (CDF) of X is given by $F_X(a) = \int_{-\infty}^{a}\int_{\mathbb{R}} \phi_{X,Y}(x,y)dy\:dx$

2. Summer days in Mannheim are hot with probability 0.8 and sunny with probability 0.5. Summer days that are both hot and sunny occur with probability 0.4. What is the probability that a randomly selected summer day will be hot or sunny?

0.7

0.85

0.9

3. Consider a probability measure $\mathcal P: \matcal A \rightarrow [0,1]$ . Which of the following properties does not hold?

For two disjoint events A and B, $P(A \cup B) = P(A) + P(B)$

For the sample space $\Omega: P(\Omega) = 1$

$P(\emptyset) \in (0,1]$

The event space $\mathcal A$ contains subsets of the sample space $\Omega$

4. The events A and B are stochastically independent. Which of the following properties are true?

$P(A \cap B) = P(A) P(B)$

$P(A \cup B) = 1 - ((1-P(A))*(1-P(B))$

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

$P(A |B) = P(B)$

5. For thwo random variables X and Y that are not independent, which of the following is wrong?

$Var(X) = \mathbb{E}(X^2) if \mathbb{E}(X) = 0$

For two scalars a and b, $Var(aX+b) = a Var(x)+b$

$Cov(X,Y) = \mathbb{E}(X,Y)-\mathbb{E}(X)\mathbb{E}(Y)$

$Var(X+Y) = Var(X) + Var(Y) + 2 Cov(X,Y)$

Question 1 of 5