GATE CE
Engineering Mathematics
Probability and Statistics
Previous Years Questions

## Marks 1

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...
The number of parameters in the univariate exponential and Gaussian distributions, respectively, are
$$X$$ 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 fol...
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 ...
Type $${\rm I}{\rm I}$$ error in hypothesis testing is
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 pro...
The annual precipitation data of a city is normally distributed with mean and standard deviation as $$1000$$mm and $$200$$ mm, respectively. The proba...
Two coins are simultaneously tossed. The probability of two heads simultaneously appearing
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 $$\,\,\,\sum x = 6,\... If the standard deviation of the speed of vehicles in a highway is$$8.8$$kmph and the mean speed of the vehicles is$$33$$kmph, the coefficient of... A box contains$$10$$screws,$$3$$of which are defective. Two screws are drawn at random with replacement. The probability that none of the two scre... ## Marks 2 For the function$$\,f\left( x \right) = a + bx,0 \le x \le 1,\,\,$$to be a valid probability density function, which one of the following statements... If$$f(x)$$and$$g(x)$$are two probability density functions,$$$f\left( x \right) = \left\{ {\matrix{ {{x \over a} + 1} & : & { - a \le... The probability density function of a random variable, $$x$$ is $$\matrix{ {f\left( x \right) = {x \over 4}\left( {4 - {x^2}} \right)} & {for... Four cards are randomly selected from a pack of$$52$$cards. If the first two cards are kings, what is the probability that the third card is a king? Consider the following probability mass function (p.m.f) of a random variable$$X.$p\left( {x,q} \right) = \left\{ {\matrix{ q & {if\,X =...
The probability density function of evaporation $$E$$ on any day during a year in a watershed is given by $$f\left( E \right) = \left\{ {\matrix{ ... A traffic office imposes on an average$$5$$number of penalties daily on traffic violators. Assume that the number of penalties on different days is... An observer counts$$240$$veh/h at a specific highway location. Assume that the vehicle arrival at the location is Poisson distributed, the probabilit... If$$\left\{ x \right\}$$is a continuous, real valued random variable defined over the interval$$\left( { - \infty ,\,\, \pm \infty } \right)$$and ... Find the value of$$\lambda $$such that the function$$f(x)$$is a valid probability density function ________. Is an experiment, positive and negative values are equally likely to occur. The probability of obtaining at most one negative value in five trials is There are two containers with one containing$$4$$red and$$3$$green balls and the other containing$$3$$blue balls and$$4$$green balls. One ball... The standard normal probability function can be approximated as$$\$F\left( {{X_N}} \right) = {1 \over {1 + \exp \left( { - 1.7255{X_N}{{\left| {{X_N}...
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...
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