Multiple Choice Exercise

*** Choose one answer and click [Submit] button

3.1 Which one is an appropriate function as a probability distribution function?

\(f(x) = \frac{x}{5} \;\;(x=-1,0,1,2,3)\)
\(f(x) = \frac{5-x}{14} \;\;(x=0,1,2,3)\)
\(f(x) = x^2 - 1 \;\;(x=0,1,2,3) \)
\(f(x) = x^3 - x^2 + 5 \;\;(x=0,1,2,3)\)

        

3.2 Which one is not the property of a probability distribution function?

\(f(x) \ge 0\)
\(\int_{-∞}^{∞} f(x)dx = 1 \)
\(P(a \le X \le b) = \int_a ^b f(x) dx\)
\(\int_{-∞}^{∞} x f(x)dx = 1 \)

        

3.3 If a random variable \(X\) has values 0, 1, 2, 3 and its probability distribution function is f(x) = \(\frac{x}{6}\), what is the mean of the random variable \(X\)?

\(\frac{5}{3}\)
\(\frac{6}{3}\)
\(\frac{7}{3}\)
\(\frac{8}{3}\)

        

3.4 If a continuous random variable \(X\) has the probability distribution function \(f(x)\) and it is defined on all real number, which one of the followings is not correct?

\(f(x) \ge 0\)
\(\int_{-∞}^{∞} f(x)dx = 1 \)
\(P(a \le X \le b) = \int_a ^b f(x) dx\)
\(\int_{0}^{∞} x f(x)dx = \frac{1}{2} \)

        

3.5 If a probability distribution function of a random variable \(X\) is as follows, what are the mean \(E(X)\) and standard deviation \(σ(X)\)?

\(X\)	0	1	2	Total
\(P(X=x)\)	\(\frac{3}{10}\)	\(\frac{6}{10}\)	\(\frac{1}{10}\)	1

E(X)=0.8, σ(X)=0.6
E(X)=0.6, σ(X)=0.8
E(X)=0.7, σ(X)=0.1
E(X)=0.1, σ(X)=0.7

        

3.6 Let the random variable X be the number of dots, 1, 2, ... , 6, which are appeared on the top when you roll a dice. Find the mean of the random variable \(X\).

2
2.5
3
3.5

        

3.7 Let the random variable \(X\) be the number of dots when you roll a dice. What is the expected value of the squared value of each \(X\)?

\(\frac{23}{6}\)
\(\frac{41}{6}\)
\(\frac{53}{6}\)
\(\frac{91}{6}\)

        

3.8 If a random variable X has \(E(X)\)=5 and \(E(X^2)\)=25, what is the variance of \(X\), \(V(X)\)?

5
0
5
25

        

3.9 If a random variable \(X\) is multiplied by 5 for each value of \(X\), the standard deviation becomes what?

5 times
1/5 times
10 times
1/10 times

        

3.10 Midterm scores of a math class have the mean 24 and standard deviation 3. If each midterm score is multiplied by 2 and add 10 points, what are the mean and standard deviation of the new midterm scores?

24, 3
48, 3
48, 6
58, 6

        

3.11 If a random variable \(X\) has the mean of 20, what is the mean of \(Y=2X+3\)?

20
40
23
43

        

3.12 If a random variable \(X\) has the variance of 2, what is the variance of \(Y=2X+3\)?

2
3
8
18

        

3.13 Which one the followings is wrong?

E(aX＋b)=aE(X)＋b
E(X＋Y)=E(X)＋E(Y)
V(aX＋b)=\(a^2\)V(X)
V(aX＋b)=\(a^2\)V(X)+b

        

3.14 What is the mean and variance of binomial distribution \(B(100, \frac{1}{6})\)?

20, 4
100, 16
100, 4
20, 16

        

3.15 What is the variance of the binomial distribution which has the distribution function as \(f(x) = {}_4 C_x (0.2)^x (1-0.2)^{4-x}\)?

0.16
0.8
0.32
0.64

        

3.16 There are 10 multiple choice questions where there is one correct answer among four possible answers. If a person chooses one answer randomly in each question, what is the probability that he would choose four correct answers?

0.0162
0.0487
0.1460
0.2050

        

3.17 If a normal distribution has the mean and standard deviation , what is the probability of the interval \(\mu ± 3 \sigma \)?

99.73% approximately
97.73% approximately
95.73% approximately
68.73% approximately

        

3.18 The entrance examination scores of a company follows a normal distribution with the mean 400 and standard deviation 50. What is the probability that a score belongs to the interval between 450 and 500 ?

80% approximately
50% approximately
30% approximately
14% approximately