MIDTERM TEST - 2025.1

Course: MTH210 - Applied Statistics
Time duration: 90 minutes
Exam number: 1

I. Single choice questions (Students need to find exactly one correct answer)

Question 1. There are 3 yellow balls, 4 green balls and 5 black balls in an urn. Four balls are drawn one by one without replacement from the urn. The probability that the first ball is black, the second ball is yellow, the third ball is green, and the fourth ball is yellow is

• $1/88$
• $1/99$
• $1/77$
• $1/66$

Question 2. The probability mass function (PMF) of a certain discrete random variable $X$ is given by $p_X(x) = P[X=x] = c(x+1)(4-x)$ for $x = 1, 2, 3$; and $p_X(x) = 0$ otherwise. The constant $c$ is

• $1/12$
• $1/10$
• $1/16$
• $1/14$

Question 3. I roll a six-sided die repeatedly until a number larger than 4 is observed. If $N$ is the total number of times that I roll the die, the probability $P[N=5]$ is

• $32/729$
• $16/729$
• $32/243$
• $16/243$

Question 4. A certain electric system contains 10 components. Suppose that the probability that each component will fail is 0.25 and that the components fail independently of each other. Given that at least one of the components has failed, the probability that there are exactly three failed components is

• $0.16522$
• $0.26522$
• $0.36522$
• $0.06522$

Question 5. In a random experiment, let $A$, $B$ and $C$ be three independent events such that $P[A] = 0.4$, $P[B] = 0.6$ and $P[C] = 0.3$. Then, $P[A \cup B \cup C]$ equals

• $0.168$
• $0.072$
• $0.928$
• $0.832$

Question 6. A company needs to design a decorative lighting system by installing 15 light bulbs with 3 types of light bulbs yellow, red, and blue. If we only care about the number of bulbs of each color type in this system, the number of possible designs is

• $105$
• $153$
• $120$
• $136$

Question 7. A continuous random variable $X$ has a PDF $f_X(x) = cx^2(2-x)$ if $x \in [0, 2]$, and $f_X(x) = 0$ if $x \notin [0, 2]$. The constant $c$ is

• $1/4$
• $3/4$
• $4/3$
• $4$

Question 8. In a particular airport, 6 radar stations operate independently of each other. The probability that a single radar station will detect an arriving airplane is 0.55. The probability that an arriving airplane will be detected by at most two radar stations is

• $0.0692$
• $0.24696$
• $0.52526$
• $0.25526$

II. Multiple choice questions (Students need to find ALL correct answers)

Question 9. Traffic engineers have coordinated the timing of two traffic lights to encourage a run of green lights. In particular, the timing was designed so that with probability 0.8 a driver will find the second light to have the same color as the first. Assuming the first light is equally likely to be red or green, what is the probability $P[G_2]$ that the second light is green? and what is $P[G_1|R_2]$, the conditional probability of a green first light given a red second light?

• $P[G_1|R_2] = 0.5$
• $P[G_1|R_2] = 0.2$
• $P[G_1|R_2] = 0.1$
• $P[G_2] = 0.1$
• $P[G_2] = 0.4$
• $P[G_2] = 0.5$

Question 10. The discrete random variable $X$ has the following probability distribution

Which of the following statements are correct?

• $Var[X] = 1.39$
• $Var[X] = 8.61$
• $Var[X] = 3.1$
• $E[X] = 1.9$
• $E[X] = 2.1$
• $E[X] = 2.2$

Question 11. The length of time in minutes that a customer queues in a bank is a random variable $T$ with PDF $f_T(t) = \frac{1}{144}(36 - t^2)$ if $t \in [0, 6]$, and $f_T(t) = 0$ if $t \notin [0, 6]$. Which of the following statements are correct?

• $P[T > 3] = 0.4125$
• $P[T > 3] = 0.6875$
• $P[T > 3] = 0.3125$
• $E[T^2] = 3.6$
• $E[T^2] = 2.25$
• $E[T^2] = 7.2$

Question 12. In a certain factory, three machines $M_1$, $M_2$, and $M_3$, make 20%, 30%, and 50%, respectively, of the products. The factory's record shows that 2%, 4% and 6% of the products made by machines $M_1$, $M_2$, and $M_3$, respectively, are defective. A finished product is selected at random from the factory. Consider two events: $A$="the selected product is defective", and $B_3$="the selected product is made by machine $M_3$". Which of the following statements are correct?

• $P[A] = 0.0046$
• $P[A] = 0.46$
• $P[A] = 0.046$
• $P[B_3|A] = 2/23$
• $P[B_3|A] = 15/23$
• $P[B_3|A] = 6/23$

III. Constructed response questions. Answer the questions completely and clearly.

Question 13. The duration of a telephone call in minutes can be modeled as a continuous random variable $T$ with PDF $f_T(t) = \frac{1}{486}(81 - t^2)$ if $t \in [0, 9]$, and $f_T(t) = 0$ if $t \notin [0, 9]$. The phone company charges 10000 VND per minute for telephone calls. What is the expected revenue per call for the phone company?

Question 14. The CDF of a continuous random variable $X$ is given by $F_X(x) = 1 - e^{-x}$ if $x \ge 0$, and $F_X(x) = 0$ if $x < 0$. Write the CDF $F_Y(y)$ of $Y = -3X + 5$.

Question 15. A local fire station receives on average $\lambda$ rescue telephone calls per day. Assume that the number of these calls in a day has a Poisson distribution. The probability that the local fire station receives at least one rescue telephone call in a day is $0.713495$. Find the parameter $\lambda$ and the probability that the local fire station will receive fewer than three rescue telephone calls over the next four days.