21. 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
22. 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
23. Consider the finite sequence of random values X = [x1, x2, ... , xn]. Let μx be the mean and σx be the standard deviation of X. Let another finite sequence Y of equal length be derived from this as yi = a ∗ xi + b, where a and b are positive constant. Let μy be the mean and σy be the standard deviation of this sequence. Which one of the following statements INCORRECT?
24. Two fair dice are rolled and the sum r of the numbers turned up is considered
25. Two coins R and S are tossed. The 4 joint events HRHS, TRTS, HRTS, TRHS 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?
26. 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 $$\int_{ - \infty }^{\text{a}} {\frac{1}{{\sqrt {2\pi } * {\text{b}}}}{{\text{e}}^{\frac{{ - 1}}{2}{{\left( {\frac{{{\text{x}} - {\text{a}}}}{{\text{b}}}} \right)}^2}}}{\text{dx}}} $$
$${\text{f}}\left( {\text{x}} \right) = \frac{1}{{\sqrt {2\pi } * {\text{b}}}}{{\text{e}}^{\frac{{ - 1}}{2}{{\left( {\frac{{{\text{x}} - {\text{a}}}}{{\text{b}}}} \right)}^2}}}$$ where 'a' and 'b' are the statistical attributes of the random variable {x}. The value of the integral $$\int_{ - \infty }^{\text{a}} {\frac{1}{{\sqrt {2\pi } * {\text{b}}}}{{\text{e}}^{\frac{{ - 1}}{2}{{\left( {\frac{{{\text{x}} - {\text{a}}}}{{\text{b}}}} \right)}^2}}}{\text{dx}}} $$
27. Type II error in hypothesis testing is
28. 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
29. A six-faced fair dice is rolled five times. The probability (in%) of obtaining "ONE" at least four times is
30. 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
The value of K is
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