Probability and Statistics MCQs Question and Answer in Engineering Maths

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 fair dice is tossed two times. The probability that the second toss results in a value that is higher than the first toss is

Consider the finite sequence of random values X = [x₁, x₂, ... , x_n]. Let μ_x be the mean and σ_x be the standard deviation of X. Let another finite sequence Y of equal length be derived from this as y_i = a ∗ x_i + b, where a and b are positive constant. Let μ_y be the mean and σ_y be the standard deviation of this sequence. Which one of the following statements INCORRECT?

Two fair dice are rolled and the sum r of the numbers turned up is considered

Two coins R and S are tossed. The 4 joint events H_RH_S, T_RT_S, H_RT_S, T_RH_S have probabilities 0.28, 0.18, 0.30, 0.24, respectively, where H represents head and T represents tail. Which one of the following is TRUE?

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:
$${\text{f}}\left( {\text{x}} \right) = \frac{1}{{\sqrt {2\pi } * {\text{b}}}}{{\text{e}}^{\frac{{ - 1}}{2}{{\left( {\frac{{{\text{x}} - {\text{a}}}}{{\text{b}}}} \right)}^2}}}$$     where 'a' and 'b' are the statistical attributes of the random variable {x}. The value of the integral $$\int_{ - \infty }^{\text{a}} {\frac{1}{{\sqrt {2\pi } * {\text{b}}}}{{\text{e}}^{\frac{{ - 1}}{2}{{\left( {\frac{{{\text{x}} - {\text{a}}}}{{\text{b}}}} \right)}^2}}}{\text{dx}}} $$

Type II error in hypothesis testing is

A random variable X has a probability density function, \[{\text{f}}\left( {\text{x}} \right) = \left\{ {\begin{array}{*{20}{c}} {{\text{k}}{{\text{x}}^{\text{n}}}{{\text{e}}^{ - {\text{x}}}};}&{{\text{x}} \geqslant {\text{0}}} \\ {0;}&{{\text{otherwise}}} \end{array}} \right.\]      (n is an integer) with mean 3. The values of {k, n} are

A six-faced fair dice is rolled five times. The probability (in%) of obtaining "ONE" at least four times is

A probability density function is of the form $${\text{p}}\left( {\text{x}} \right) = {\text{K}}{{\text{e}}^{ - \alpha \left| x \right|}},\,{\text{x}} \in \left( { - \infty ,\,\infty } \right).$$
The value of K is