Probability and Statistics · Engineering Mathematics · GATE CE

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Marks 1

1
Given that $A$ and $B$ are not null sets, which of the following statements regarding probability is/are CORRECT?
GATE CE 2025 Set 2
2

$X$ is the random variable that can take any one of the values, $0,1,7,11$ and 12 . The probability mass function for $X$ is

$$ \begin{aligned} & \mathrm{P}(X=0)=0.4 ; \mathrm{P}(X=1)=0.3 ; \mathrm{P}(X=7)=0.1 ; \\ & \mathrm{P}(X=11)=0.1 ; \mathrm{P}(X=12)=0.1 \end{aligned} $$

Then, the variance of $X$ is

GATE CE 2025 Set 1
3

The probability that a student passes only in Mathematics is $\frac{1}{3}$. The probability that the student passes only in English is $\frac{4}{9}$. The probability that the student passes in both of these subjects is $\frac{1}{6}$. The probability that the student will pass in at least one of these two subjects is

GATE CE 2024 Set 1
4

Which of the following probability distribution functions (PDFs) has the mean greater than the median?

GATE CE 2023 Set 2 Engineering Mathematics - Probability and Statistics Question 6 English
GATE CE 2023 Set 2
5

A remote village has exactly 1000 vehicles with sequential registration numbers starting from 1000. Out of the total vehicles, 30% are without pollution clearance certificate. Further, even- and odd-numbered vehicles are operated on even- and odd-numbered dates, respectively.

If 100 vehicles are chosen at random on an even-numbered date, the number of vehicles expected without pollution clearance certificate is ________.

GATE CE 2023 Set 2
6
The probabilities of occurrences of two independent events A and B are 0.5 and 0.8, respectively. What is the probability of occurrence of at least A or B (rounded off to one decimal place)? ________
GATE CE 2023 Set 1
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 _____.
GATE CE 2017 Set 2
8
The number of parameters in the univariate exponential and Gaussian distributions, respectively, are
GATE CE 2017 Set 1
9
$$X$$ and $$Y$$ are two random independent events. It is known that $$P(X)=0.40$$ and $$\,P\left( {X \cup {Y^C}} \right) = 0.7.$$ Which one of the following is the value of $$P\left( {X \cup Y} \right)$$ $$?$$
GATE CE 2016 Set 2
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 __________.
GATE CE 2016 Set 2
11
Type $${\rm I}{\rm I}$$ error in hypothesis testing is
GATE CE 2016 Set 1
12
A fair (unbiased) coin was tossed four times in succession and resulted in the following outcomes; (i) Head, (ii) Head, (III) Head, (iv) Head. The probability of obtaining a ''Tail'' when the coin is tossed again is
GATE CE 2014 Set 2
13
The annual precipitation data of a city is normally distributed with mean and standard deviation as $$1000$$mm and $$200$$ mm, respectively. The probability that the annual precipitation will be more than $$1200$$ mm is
GATE CE 2012
14
Two coins are simultaneously tossed. The probability of two heads simultaneously appearing
GATE CE 2010
15
Three values of $$x$$ and $$y$$ are to be fitted in a straight line in the form $$y=a+bx$$ by the method of least squares. Given $$\,\,\,\sum x = 6,\,\,\sum y = 21,\,\,\sum {{x^2} = 14,\,\,\sum {xy} = 46,\,\,\,\,} $$ the values of $$a$$ and $$b$$ are respectively
GATE CE 2008
16
If the standard deviation of the speed of vehicles in a highway is $$8.8$$ kmph and the mean speed of the vehicles is $$33$$ kmph, the coefficient of variation in speed is
GATE CE 2007
17
A box contains $$10$$ screws, $$3$$ of which are defective. Two screws are drawn at random with replacement. The probability that none of the two screws is defective will be
GATE CE 2003

Marks 2

1

Consider a discrete random variable $X$ whose probabilities are given below. The standard deviation of the random variable is _________ (round off to one decimal place).

$$ \begin{array}{|c|c|c|c|c|} \hline x_1 & 1 & 2 & 3 & 4 \\ \hline P\left(X=x_i\right) & 0.3 & 0.1 & 0.3 & 0.3 \\ \hline \end{array} $$

GATE CE 2025 Set 2
2

A one-way, single lane road has traffic that consists of $30 \%$ trucks and $70 \%$ cars. The speed of trucks (in km/h) is a uniform random variable on the interval ( 30,60 ), and the speed of cars (in km/h) is a uniform random variable on the interval $(40,80)$. The speed limit on the road is $50 \mathrm{~km} / \mathrm{h}$. The percentage of vehicles that exceed the speed limit is ________ (rounded off to 1 decimal place).

Note: $X$ is a uniform random variable on the interval ( $\alpha, \beta$ ), if its probability density function is given by

$$ f(x)= \begin{cases}\frac{1}{\beta-\alpha} & \alpha < x < \beta \\ 0 & \text { otherwise }\end{cases} $$

GATE CE 2025 Set 1
3

In a sample of 100 heart patients, each patient has 80% chance of having a heart attack without medicine X. It is clinically known that medicine X reduces the probability of having a heart attack by 50%. Medicine X is taken by 50 of these 100 patients. The probability that a randomly selected patient, out of the 100 patients, takes medicine X and has a heart attack is

GATE CE 2024 Set 2
4

The return period of a large earthquake for a given region is 200 years. Assuming that earthquake occurrence follows Poisson’s distribution, the probability that it will be exceeded at least once in 50 years is ______________ % (rounded off to the nearest integer).

GATE CE 2024 Set 1
5

A pair of six-faced dice is rolled twice. 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).

GATE CE 2022 Set 2
6
For the function $$\,f\left( x \right) = a + bx,0 \le x \le 1,\,\,$$ to be a valid probability density function, which one of the following statements is correct?
GATE CE 2017 Set 1
7
If $$f(x)$$ and $$g(x)$$ are two probability density functions, $$$f\left( x \right) = \left\{ {\matrix{ {{x \over a} + 1} & : & { - a \le x < 0} \cr { - {x \over a} + 1} & : & {0 \le x \le a} \cr 0 & : & {otherwise} \cr } } \right.$$$ $$$g\left( x \right) = \left\{ {\matrix{ { - {x \over a}} & : & { - a \le x < 0} \cr {{x \over a}} & : & {0 \le x \le a} \cr 0 & : & {otherwise} \cr } } \right.$$$

Which of the following statements is true?

GATE CE 2016 Set 2
8
Consider the following probability mass function (p.m.f) of a random variable $$X.$$ $$$p\left( {x,q} \right) = \left\{ {\matrix{ q & {if\,X = 0} \cr {1 - q} & {if\,X = 1} \cr 0 & {otherwise} \cr } } \right.$$$
$$q=0.4,$$ the variance of $$X$$ is _______.
GATE CE 2015 Set 1
9
The probability density function of a random variable, $$x$$ is $$$\matrix{ {f\left( x \right) = {x \over 4}\left( {4 - {x^2}} \right)} & {for\,\,0 \le x \le 2 = 0} \cr { = 0} & {otherwise} \cr } $$$
The mean, $${\mu _x}$$ of the random variable is __________.
GATE CE 2015 Set 2
10
Four cards are randomly selected from a pack of $$52$$ cards. If the first two cards are kings, what is the probability that the third card is a king?
GATE CE 2015 Set 2
11
If $$\left\{ x \right\}$$ is a continuous, real valued random variable defined over the interval $$\left( { - \infty ,\,\, \pm \infty } \right)$$ and its occurrence is defined by the density function given as: $$f\left( x \right) = {1 \over {\sqrt {2\pi * b} }}{e^{ - {1 \over 2}{{\left( {{{x - a} \over b}} \right)}^2}}}$$ where $$'a'$$ and $$'b'$$ are the statistical attributes of the random variable $$\left\{ x \right\}$$. The value of the integral $$\int\limits_{ - \infty }^a {{1 \over {\sqrt {2\pi * b} }}{e^{ - {1 \over 2}{{\left( {{{x - a} \over b}} \right)}^2}}}} dx\,\,\,$$ is
GATE CE 2014 Set 2
12
An observer counts $$240$$veh/h at a specific highway location. Assume that the vehicle arrival at the location is Poisson distributed, the probability of having one vehicle arriving over a $$30$$-second time interval is _______.
GATE CE 2014 Set 2
13
The probability density function of evaporation $$E$$ on any day during a year in a watershed is given by $$$f\left( E \right) = \left\{ {\matrix{ {{1 \over 5}} & {0 \le E \le 5\,mm/day} \cr 0 & {otherwise} \cr } } \right.$$$
The probability that $$E$$ lies in between $$2$$ and $$4$$ $$mm/day$$ in the watershed is (in decimal) _______.
GATE CE 2014 Set 1
14
A traffic office imposes on an average $$5$$ number of penalties daily on traffic violators. Assume that the number of penalties on different days is independent and follows a Poisson distribution. The probability that there will be less than $$4$$ penalties in a day is ________.
GATE CE 2014 Set 1
15
Find the value of $$\lambda $$ such that the function $$f(x)$$ is a valid probability density function ________.
GATE CE 2013
16
Is an experiment, positive and negative values are equally likely to occur. The probability of obtaining at most one negative value in five trials is
GATE CE 2012
17
There are two containers with one containing $$4$$ red and $$3$$ green balls and the other containing $$3$$ blue balls and $$4$$ green balls. One ball is drawn at random from each container. The probability that one of the balls is red and the other is blue will be ___________.
GATE CE 2011
18
The standard normal probability function can be approximated as $$$F\left( {{X_N}} \right) = {1 \over {1 + \exp \left( { - 1.7255{X_N}{{\left| {{X_N}} \right|}^{0.12}}} \right)}}\,\,\,$$$
$$\,{X_N} = $$ standard normal deviate. If mean and standard deviation of annual precipitation are $$102$$ cm and $$27$$ cm respectively, the probability that the annual precipitation will be $$b/w$$ $$90$$ cm and $$102$$ cm is
GATE CE 2009
19
A hydraulic structure has four gates which operate independently. The probability of failure of each gate is $$0.2.$$ Given that gate $$1$$ has failed, the probability that both gates $$2$$ and $$3$$ will fail is
GATE CE 2004
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