Question

1. (a) For each one of the statements below say whether the statement is true or false explaining your answer. A and B are events with probabilities in (0,1). i. If $P(A) = P(B)$, then it holds that $P(A|B) = P(B|A)$. ii. It holds that: $P(A|B) + P(A|B^c) > P(A)$ where $^c$ denotes a complementary event. iii. If $P(A) < P(B)$, then A is a subset of B. iv. For any discrete random variable X, it holds that $P(X = E(X)) > 0$. v. If T is an unbiased estimator of the parameter $\theta > 0$, then $1/T$ is an unbiased estimator of $1/\theta$. (10 marks) (b) Let $Z_1$ and $Z_2$ be two independent and normally distributed random variables, both with mean 0 and variance 1. Calculate: $P(3Z_1 - Z_2 > 6 | 3Z_1 - Z_2 > 5)$. (6 marks)

          1. (a) For each one of the statements below say whether the statement is true or false explaining your answer. A and B are events with probabilities in (0,1).
i. If $P(A) = P(B)$, then it holds that $P(A|B) = P(B|A)$.
ii. It holds that:
$P(A|B) + P(A|B^c) > P(A)$
where $^c$ denotes a complementary event.
iii. If $P(A) < P(B)$, then A is a subset of B.
iv. For any discrete random variable X, it holds that $P(X = E(X)) > 0$.
v. If T is an unbiased estimator of the parameter $\theta > 0$, then $1/T$ is an unbiased estimator of $1/\theta$.
(10 marks)
(b) Let $Z_1$ and $Z_2$ be two independent and normally distributed random variables, both with mean 0 and variance 1. Calculate:
$P(3Z_1 - Z_2 > 6 | 3Z_1 - Z_2 > 5)$.
(6 marks)

1. (a) For each one of the statements below say whether the statement is true or false explaining your answer. A and B are events with probabilities in (0,1).
i. If P(A) = P(B), then it holds that P(A|B) = P(B|A).
ii. It holds that:
P(A|B) + P(A|B^c) > P(A)
where ^c denotes a complementary event.
iii. If P(A) < P(B), then A is a subset of B.
iv. For any discrete random variable X, it holds that P(X = E(X)) > 0.
v. If T is an unbiased estimator of the parameter θ > 0, then 1/T is an unbiased estimator of 1/θ.
(10 marks)
(b) Let Z1 and Z2 be two independent and normally distributed random variables, both with mean 0 and variance 1. Calculate:
P(3Z1 - Z2 > 6 | 3Z1 - Z2 > 5).
(6 marks)

Added by Sarah M.

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