Probability and Statistics

Question 1
A pair of six-faced dice is rolled thrice. The probability that the sum of the outcomes in each roll equals 4 in exactly two of the three attempts is ______. (round off to three decimal places)
A
0.045
B
0.078
C
0.018
D
0.025
GATE CE 2022 SET-2   Engineering Mathematics
Question 1 Explanation: 
Event, E = {(1, 3)(3, 1)(2, 2)}
n(E) = 3
n(S) = 36
p=P(E)=\frac{3}{36}=\frac{1}{12}
q=P(\bar{E})=1-\frac{1}{12}=\frac{11}{12}
P(x)=3C_2(p^2)(q^1)=3 \times \left (\frac{1}{12} \right )^2\left (\frac{11}{12} \right )=0.02
Question 2
Match the following attributes of a city with the appropriate scale of measurements.
\begin{array}{|l|l|}\hline \text{Attribute}&\text{Scale of measurement} \\ \hline \text{(P) Average temperature } (^{\circ}C)\text{ of a city} & \text{((I) Interval} \\ \hline \text{(Q) Name of a city} & \text{(II) Ordinal}\\ \hline \text{(R) Population density of a city}& \text{(III) Nominal}\\ \hline \text{(S) Ranking of a city based on ease of business}& \text{(IV) Ratio}\\ \hline \end{array}
Which one of the following combinations is correct?
A
(P)-(I), (Q)-(III), (R)-(IV), (S)-(II)
B
(P)-(II), (Q)-(I), (R)-(IV), (S)-(III)
C
(P)-(II), (Q)-(III), (R)-(IV), (S)-(I)
D
(P)-(I), (Q)-(II), (R)-(III), (S)-(IV)
GATE CE 2022 SET-2   Engineering Mathematics
Question 2 Explanation: 
Meaning of
Nominal -> a name or term
Ordinal -> in an ordered sequence
Ratio -> quantitative relation between two things
Interval -> indicates average of a range
Question 3
A set of observations of independent variable (x) and the corresponding dependent variable (y) is given below.
\begin{array}{|c|c|c|c|c|} \hline x&5&2&4&3 \\ \hline y&16&10&13&12\\ \hline \end{array}
Based on the data, the coefficient a of the linear regression model
y = a + bx
is estimated as 6.1.
The coefficient b is ______________ . (round off to one decimal place)
A
6.1
B
1.9
C
2.2
D
3.6
GATE CE 2022 SET-1   Engineering Mathematics
Question 3 Explanation: 
We know that, normal equation for fitting of straight lines are
\begin{aligned} \Sigma y&=na+b\Sigma x\\ \Sigma xy&=a\Sigma x+b\Sigma x^2\\ n&=4\\ 51&=4a+b(14)\\ 188&=a(14)+b(54) \end{aligned}
After solving, a=6.1 and b=1.9
Question 4
The shape of the cumulative distribution function of Gaussian distribution is
A
Horizontal line
B
Straight line at 45 degree angle
C
Bell-shaped
D
S-shaped
GATE CE 2021 SET-1   Engineering Mathematics
Question 4 Explanation: 


PDF:f(x)=\frac{1}{\sigma \sqrt{2 \pi}}e^{-(x-\mu )^2/(2\sigma ^2)}
CDF:F(x)=\frac{1}{2}\left [ 1+eff\left ( \frac{x-\mu }{\sigma \sqrt{2}} \right ) \right ]
Question 5
A fair (unbiased) coin is tossed 15 times. The probability of getting exactly 8 Heads (round off to three decimal places), is ________.
A
0.523
B
0.421
C
0.196
D
0.223
GATE CE 2020 SET-2   Engineering Mathematics
Question 5 Explanation: 
P(H)=\frac{1}{2}
P(T)=\frac{1}{2}
Probability of getting exactly 8 heads out of 15 trial ={^{15}C_8}\left [ \frac{1}{2} \right ]^8 \times \left [ \frac{1}{2} \right ]^{15-8}=0.196
Question 6
The probability density function of a continuous random variable distributed uniformly between x and y (for y \gt x) is
A
\frac{1}{x-y}
B
\frac{1}{y-x}
C
x-y
D
y-x
GATE CE 2019 SET-2   Engineering Mathematics
Question 6 Explanation: 
Probability density function of uniform distribution is f(x)=\frac{1}{y-x}
Question 7
Probability (up to one decimal place) of consecutively picking 3 red balls without replacement from a box containing 5 red balls and 1 white ball is ______
A
0
B
0.25
C
0.5
D
1
GATE CE 2018 SET-2   Engineering Mathematics
Question 7 Explanation: 
Probability, \bar{P}=\frac{5}{6}\times \frac{4}{5}\times \frac{3}{4}=\frac{1}{2}=0.5
Question 8
A probability distribution with right skew is shown in the figure.

The correct statement for the probability distribution is
A
Mean is equal to mode
B
Mean is greater than median but less than mode
C
Mean is greater than median and mode
D
Mode is greater than median
GATE CE 2018 SET-2   Engineering Mathematics
Question 8 Explanation: 



t_{L}\lt t_{mean}=Curve is skew to right.
Mode \lt mean
i.e., Mean \gt median and mode
Mean is greater than the mode and the median.This is common for a distribution that is skewed to the right [i.e., bunched up toward the left and a 'tail' stretching toward the right].
Question 9
The graph of a function f(x) is shown in the figure.

For f(x) to be a valid probability density function, the value of h is
A
0.33
B
0.66
C
1
D
3
GATE CE 2018 SET-2   Engineering Mathematics
Question 9 Explanation: 
\begin{aligned} \int_{0}^{3}f\left ( x \right )dx &=1 \\ \int_{0}^{1}f\left ( x \right )dx+\int_{1}^{2}f\left ( x \right )dx+\int_{2}^{3}f\left ( x \right )dx &=1 \\ \frac{h}{2}+\frac{2h}{2+\frac{3h}{2}} &=1 \\ 6h &=2 \\ \Rightarrow\; \; h &=\frac{1}{3} \end{aligned}
Question 10
A two-faced fair coin has its faces designated as head (H) and tail (T). This coin is tossed three times in succession to record the following outcomes: H, H, H. If the coin is tossed one more time, the probability (up to one decimal place) of obtaining H again, given the previous realizations of H, H and H, would be______
A
0.25
B
0.5
C
0.75
D
0.2
GATE CE 2017 SET-2   Engineering Mathematics
There are 10 questions to complete.

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