Chapter 5 – Quiz 1 – E600 Mathematics

1. 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?

0.85

0.7

0.9

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

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

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

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

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

3. 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?

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Question 1 of 5