Probability and Statistics

Question 1
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 1 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 2
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 2 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 3
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 3 Explanation: 
Probability density function of uniform distribution is f(x)=\frac{1}{y-x}
Question 4
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 4 Explanation: 
Probability, \bar{P}=\frac{5}{6}\times \frac{4}{5}\times \frac{3}{4}=\frac{1}{2}=0.5
Question 5
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 5 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 6
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 6 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 7
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
Question 8
If f(x) and g(x) are two probability density functions,
f(x)=\left\{\begin{matrix} \frac{x}{a}+1 & :-a\leq x \lt 0\\ -\frac{x}{a}+1 & :0\leq x\leq a\\ 0& : otherwise \end{matrix}\right.

g(x)=\left\{\begin{matrix} -\frac{x}{a} & :-a\leq x \lt 0\\ \frac{x}{a} & :0\leq x\leq a\\ 0& : otherwise \end{matrix}\right.
Which one of the following statements is true?
A
Mean of f(x) and g(x) are same; Variance of f(x) and g(x) are same.
B
Mean of f(x) and g(x) are same; Variance of f(x) and g(x) are different.
C
Mean of f(x) and g(x) are different; Variance of f(x) and g(x) are same.
D
Mean of f(x) and g(x) are different; Variance of f(x) and g(x) are different.
GATE CE 2016 SET-2   Engineering Mathematics
Question 8 Explanation: 
Mean of f(x) is E(x)
=\int_{-a}^{0}x\left ( \frac{x}{a}+1 \right )dx+\int_{0}^{a}x\left ( \frac{-x}{a}+1 \right )dx
=\left ( \frac{x^{3}}{3a}+\frac{x^{2}}{2} \right )_{-a}^{0}+\left ( \frac{-x^{3}}{3a}+\frac{x^{2}}{2} \right )_{0}^{a}=0
Variance of f(x) is E\left ( x \right )^{2}-\left \{ E\left ( x \right )^{2} \right \} where,
E\left ( x \right )^{2}=\int_{-a}^{0}x^{2}\left ( \frac{x}{a}+1 \right )dx+\int_{0}^{a}x^{2}\left ( \frac{-x}{a}+1 \right )dx
=\left ( \frac{x^{4}}{4a}+\frac{x^{3}}{3} \right )_{-a}^{0}+\left ( \frac{-x^{4}}{4a}+\frac{x^{3}}{3} \right )_{0}^{a}=\frac{a^{3}}{6}
\Rightarrow \; \; Variance is \frac{a^{3}}{6}
Next, mean of g(x) is E(x)
=\int_{a}^{0}s\left ( \frac{-x}{a} \right )dx+\int_{0}^{a}x\left ( \frac{x}{a} \right )dx=0
Variance of g(x) is E\left ( x^{2} \right )-\left \{ E\left ( x \right )^{2} \right \}, where,
E\left ( x^{2} \right )=\int_{-a}^{0}x^{2}\left ( \frac{-x}{a} \right )dx+\int_{0}^{a}x^{2}\left ( \frac{x}{a} \right )dx=\frac{a^{3}}{2}
\Rightarrow \; \; Variance is \frac{a^{3}}{2}
\therefore Mean of f(x) and g(x) are same but variance of f(x) and g(x) are different.
Question 9
X and Y are two random independent events. It is known that P(X)=0.40 and P(X\cup Y^{C})=0.7. Which one of the following is the value of P(X\cup Y) ?
A
0.7
B
0.5
C
0.4
D
0.3
GATE CE 2016 SET-2   Engineering Mathematics
Question 9 Explanation: 
\; \; \; \; P\left ( X\: \cup \: Y^{c} \right )=0.7
\Rightarrow \; \; P\left ( X \right )+P\left ( Y^{c} \right )-P\left ( X \right )P\left ( Y^{c} \right )=0.7
(Since X, Y are independent events)
\Rightarrow \; \; P\left ( X \right )+1-P\left ( Y \right )-P\left ( X \right )\left \{ 1-P\left ( Y \right ) \right \}=0
\Rightarrow \; \; P\left ( X \right )-P\left ( X\: \cap \: Y \right )=0.3\; \; \; \; \; \; ...\left ( i \right )
\; \; \; \; P\left ( X\: \cup \: Y \right )=P\left ( X \right )+P\left ( Y \right )-P\left ( X\: \cap \: Y \right )
\; \; \; \; =0.4+0.3=0.7
Question 10
The spot speeds (expressed in km/hr) observed at a road section are 66, 62, 45, 79, 32, 51, 56, 60, 53, and 49. The median speed (expressed in km/hr) is ________.
(Note: answer with one decimal accuracy)
A
54.5
B
51.5
C
53.5
D
56
GATE CE 2016 SET-2   Engineering Mathematics
Question 10 Explanation: 
Median speed is the speed at the middle value in series of spot speeds that are arranged in ascending order. 50% of speed values will be greater than the median 50% will be less than the median.
Ascending order order of spot speed studies are 32, 39, 45, 51, 53, 56, 60, 62, 66, 79
Median speed=\frac{53+56}{2}=54.5 km/hr
There are 10 questions to complete.

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