Probability & Stats (CE)

$X$ 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 ; \\ & \ma...

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 &...

Given that $A$ and $B$ are not null sets, which of the following statements regarding probability is/are CORRECT?

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...

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 $...

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 ta...

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 _________...

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)? ________

There are 4 red, 5 green, and 6 blue balls inside a box. If 𝑁 number of balls are picked simultaneously, what is the smallest value of 𝑁 that guarantees there will be at least two balls of the same colour? One cannot s...

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...

A survey of 450 students about their subjects of interest resulted in the following outcome. $$\bullet$$ 150 students are interested in Mathematics. $$\bullet$$ 200 students are interested in Physics. $$\bullet$$ 175 stu...

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).

A set of observations of independent variable (x) and the corresponding dependent variable (y) is given below. x 5 2 4 3 y 16 10 13 12 Based on the data, the coefficient a of the linear regression model y = a + bx is est...

A three-member committee has to be formed from a group of 9 people. How many such distinct committees can be formed?

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?

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 probab...

The number of parameters in the univariate exponential and Gaussian distributions, respectively, are

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...

$$X$$ 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 $$P\left( {X \cup Y} \right)$$ $$?$$

Type $${\rm I}{\rm I}$$ error in hypothesis testing is

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 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}...

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 probability of having one vehicle arriving over a $$30$$-second time interval is...

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 } } \righ...

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

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(...

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 t...

Find the value of $$\lambda $$ such that the function $$f(x)$$ is a valid probability density function ________.

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

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 is drawn at random from each container. The probability that one of t...

Two coins are simultaneously tossed. The probability of two heads simultaneously appearing

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 d...

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

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

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