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.