## 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...

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 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?

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 ...

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...

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...