Questions without answers in Probability and Statistics

The probability (in $\%$) that a storm having return period of $15$ years may occur in the next $10$ years is $\_\_\_\_$ (rounded off to two decimal places).

The age (in years) of a population is normally distributed with a mean of $36$ and standard deviation of $12$. The height (in cm) of the same population is also normally ...

Bag $\text{I}$ contains $4$ white and $6$ black balls.Bag $\text{II}$ contains $4$ white and $3$ black balls.One ball is drawn at random from any one of the two bags and ...

Given that $A$ and $B$ are not null sets, which of the following statements regarding probability is/are CORRECT?$\mathrm{P}(\mathrm{A} \cap \mathrm{B})=\mathrm{P}(\mathr...

Consider a discrete random variable $X$ whose probabilities are given below. The standard deviation of the random variable is $\_\_\_\_\_\_\_\_\_$ (round off to one decim...

​​​​​$X$ is a random variable that can take any one of the values $0,1,7,11$, and $12$ . The probability mass function for $X$ is$$\begin{array}{l}P(X=0)=0.4 ; \quad P(X=...

A one-way, single lane road has traffic that consists of $30 \%$ trucks and $70 \%$ cars. The speed of trucks (in $\mathrm{km} / \mathrm{h}$ ) is a uniform random variabl...

In a sample of $100$ heart patients, each patient has $80 \%$ chance of having a heart attack without medicine $\mathrm{X}$. It is clinically known that medicine $\mathrm...

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 wil...

The probabilities of occurrences of two independent events $\text {A}$ and $\text {B}$ are $0.5$ and $0.8,$ respectively. What is the probability of occurrence of at leas...

Match the following attributes of a city with the appropriate scale of measurements.$$\begin{array} {|l|l|} \hline \qquad \qquad \qquad \textbf{Attribute} & \textbf{ Sca...

A pair of six-faced dice is rolled thrice. The probability that the sum of the outcomes in each roll equals $4$ in exactly two of the three attempts is _______. (round of...

A set of observations of independent variable $(x)$ and the corresponding dependent variable $(y)$ is given below.$$\begin{array}{|c|c|} \hline\text{x} & 5 & 2 & 4 & 3 \\...

At a traffic intersection, cars and buses arrive randomly according to independent Poisson processes at an average rate of $4$ vehicles per hour and $2$ vehicles per hour...

The shape of the cumulative distribution function of Gaussian distribution isHorizontal lineStraight line at $45$ degree angleBell-shaped$S$-shaped

A fair (unbiased) coin is tossed $15$ times. The probability of getting exactly $8$ Heads (round off to three decimal places), is _______.

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 c...

The probability that the annual maximum flood discharge will exceed $25000 \: m^3/s$, at least once in next $5$ years is found to be $0.25$. The return period of this flo...

The probability density function of a continuous random variable distributed uniformly between $x$ and $y$ (for $y>x$) is$\dfrac{1}{x-y}$$\dfrac{1}{y-x}$$x-y$$y-x$

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...

$X$ 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 $P(X \cup Y)$?$0.7$$0.5$$0...

If $f(x)$ and $g(x)$ are two probability density functions,$f(x) = \begin{cases} \dfrac{x}{a}+1 & :-a \leq x < 0 \\ -\dfrac{x}{a}+1 & : 0 \leq x \leq a \\ 0 & :\text{oth...

Type II error in hypothesis testing isacceptance of the null hypothesis when it is false and should be rejectedrejection of the null hypothesis when it is true and should...

Probability density function of a random variable $X$ is given below$$f(x) = \begin{cases} 0.25 & \text{ if } 1 \leq x \leq 5 \\ 0 & \text{ otherwise} \end{cases}$$$P(X \...

The number of parameters in the univariate exponential and Gaussian distributions, repsectively, are$2$ and $2$$1$ and $2$$2$ and $1$$1$ and $1$

For the function $f(x)=a+bx,0\leq x\leq1,$ to be a valid probability density function,which one of the following statements is correct?$a=1,b=4$$b=0.5,b=1$$a=0,b=1$$a=1,b...

The probability density function of a random variable, $x$ is$f(x)= \begin{cases} \dfrac{x}{4} ( 4-x^2) & \text{ for } 0 \leq x \leq 2 \\ 0 & \text{ otherwise} \end{ca...

Consider the following probability mass function (p.m.f) of a random variable $X:$$$p(x,q) = \begin{cases} q & \text{ if } X=0 \\ 1 – q & \text{ if } X=1 \\ 0 & \text{oth...

If $\{x\}$ is a continuous, real valued random variable defined over the interval $(- \infty, + \infty)$ and its occurrence is defined by the density function given as: $...

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 obtainin...

The probability density function of evaporation $E$ on any day during a year in a watershed is given by$f(E) = \begin{cases} \dfrac{1}{5} & 0 \leq E \leq 5\: \text{mm/day...

Find the value of $\lambda$ such that the function $f(x)$ is a valid probability density function _______$f(x) = \begin{cases} \lambda (x-1)(2-x) & \text{for } 1 \leq x \...

A $1$-h rainfall of $10$ cm magnitude at a station has a return period of $50$ years. The probability that a $1$-h rainfall of magnitude $10$ cm or more will occur in eac...

In an experiment, positive and negative values are equally likely to occur. The probability of obtaining at most one negative value in five trials is$\dfrac{1}{32} \\$$\d...

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 ...

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 ___________

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$1/3$$2/3$$1$$3$

A probability distribution with right skew is shown in the figure.The correct statement for the probability distribution isMean is equal to modeMean is greater than media...