True or False questions.

(a) The matrix $\left[ \begin{array}{c c c | c} 1 & 0 & 0 & 0 \ 0 & 1 & 0 & 0 \ 0 & 0 & 1 & 0 \ 0 & 0 & 0 & 0 \end{array} \right]$ corresponds to a system of linear equations with unique soln $x=y=z=0$.

(a) The matrix $\left[ \begin{array}{c c c | c} 1 & 0 & 0 & 0 \ 0 & 1 & 0 & 0 \ 0 & 0 & 1 & 0 \ 0 & 0 & 0 & 0 \end{array} \right]$ corresponds to a system of linear equations with unique soln $x=y=z=0$.

(b) Given two mutually exclusive events $E$ and $F$, we have $P(E \; \text{or} \; F)$=$P(E)+P(F)$.

(c) If $E$ and $F$ are independent, events then $P(E \; \text{and} \; F)=P(E)\cdot P(F/E)$

T (d) If $I$ is the $3 \times 3$ identify matrix and $A$ is any $3*3$ matrix, then $AI=IA$.

(e) Roll a die and record the number and let $E$ and $F$ be the following events $E={ 2,4,6 }$ and $F={ 1,3,5 }$. Then the events $E$ and $F$ are independent.

2. Find the matrix product of $AB$and $BA$ if

3. Solve the system of linear equations with augmented matrix $A$ given below. Use elementary row operations to obtain the rref(reduced row echelon from)of $A$ and be precise in your answer. you should assume that the column that the column variables are $x,y,z$ in the usual order.

4. Consider an experiment where two fair dice are rolled and the sum of the two numbers are recorded. Let $X$ be the sum of the two nembers which appear face up on the dice. Find the expected calue and variance of $X$.

5. Suppose four fair die are rolled. What is the probability that at least one of the die shows either a 1 or a 2?

6. Consider the following two stage experiment. First, we draw a card from a 52 card deck. If the card face card then we flip a coin, and if it is not a face card then we roll a die. Find the probability that we end the sequence with a "6" on the die or with a "heads" on the coin.

7. Let $X$ be a normally distributed continous random variable withand

8. A washing machine manafacturer knows that 2% of its machines breaks down in the first year. Estimate the probability of at least 15 out of 1000 washes breaking down in the first year.

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