Probability and statistics
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Probability and statistics questions with answers

71 multiple-choice questions. Open any question for its answer and explanation.

  1. Two independent random variables X and Y are uniformly distributed in the interval [-1, 1]. The probability that max [X, Y] is less than $$\frac{1}{2}$$ is
  2. The standard deviation of a uniformly distributed random variable between 0 and 1 is
  3. Suppose afair six-sided die is rolled once. If the value on the die is 1, 2 or 3 the die is rolled a second time. What is the probability that the sum of total values that turn up is at least 6?
  4. The number of parameters in the univariate exponential and Gaussian distributions, respectively, are
  5. The probability density function of a continuous random variable distributed uniformly between x and y (for y > x) is
  6. If P(x) = $$\frac{1}{4}$$, P(Y) = $$\frac{1}{3}$$ and $${\text{P}}\left( {{\text{X}} \cap {\text{Y}}} \right) = \frac{1}{{12}},$$ the value of $${\text{P}}\left( {\frac{{\text{Y}}}{{\text{X}}}} \right)$$ is
  7. The probability that a communication system will have high fidelity is 0.81. The probability that the system will have both high fidelity and high selectivity is 0.18. The probability that a given system with high fidelity will have high selectivity is
  8. Probability density function of a random variable X is given below \[{\text{f}}\left( {\text{x}} \right) = \left[ {\begin{array}{*{20}{c}} {0.25}&{{\text{if }}1 \leqslant {\text{x}} \geqslant 5} \\ 0&{{\text{otherwise}}} \end{array}} \right]\,{\text{P}}\left( {{\text{X}} \leqslant 4} \right)\]
  9. A fail coin is tossed N times. The probability that head does not turn up in any of the tosses is
  10. There are 25 calculators in a box. Two of them are defective. Suppose 5 calculators are randomly picked for inspection (i.e., each has the same chance of being selected), what is the probability that only one of the defective calculators will be included in the inspection?
  11. A single dice is thrown twice. What is the probability that the sum is neither 8 nor 9?
  12. Four red balls, four green balls and four blue balls are put in a box. Three balls are pulled our of the box at random one after another without replacement. The probability that all the three balls are red is
  13. The chance of a student passing an exam is 20%. The chance of a student passing the exam and getting above 90% marks in it is 5%. Given that a student passes the examination, the probability that the student gets above 90% marks is
  14. A program consists of two modules executed sequentially. Let f1(t) and f2(t) respectively denote the probability density functions of time taken to execute the two modules. The probability density function of the overall time taken to execute the program is given by
  15. The random variable X takes on the values 1, 2 (or) 3 with probabilities $$\frac{{2 + 5{\text{P}}}}{5},\frac{{1 + 3{\text{P}}}}{5}$$ and $$\frac{{1.5 + 2{\text{P}}}}{5}$$ respectively the values of P and E(X) are respectively.
  16. There are two containers, with one containing 4 red and 3 green balls and the other containing 3 blue and 4 green balls. One ball is drawn at random from each container. The probability that one of the balls is red and the other is blue will be
  17. A fair coin is tossed three times in succession. If the first toss produces a head, then the probability of getting exactly two heads in three tosses is
  18. Marks obtained by 100 students in an examination are given in the table: Sr. No. Marks obtained No. of students 1 25 20 2 30 20 3 35 40 4 40 20 What would be mean, median and mode of marks obtained by the students?
  19. If f(x) and g(x) are two probability density functions, \[{\text{f}}\left( {\text{x}} \right) = \left\{ {\begin{array}{*{20}{c}} {\frac{{\text{x}}}{{\text{a}}} + 1}&:&{ - {\text{a}} \leqslant {\text{x}} < 0} \\ { - \frac{{\text{x}}}{{\text{a}}} + 1}&:&{0 \leqslant {\text{x}} \leqslant {\text{a}}} \\ 0&:&{{\text{otherwise}}} \end{array}} \right.;\,\,{\text{g}}\left( {\text{x}} \right) = \left\{ {\begin{array}{*{20}{c}} { - \frac{{\text{x}}}{{\text{a}}}}&:&{ - {\text{a}} \leqslant {\text{x}} < 0} \\ {\frac{{\text{x}}}{{\text{a}}}}&:&{0 \leqslant {\text{x}} \leqslant {\text{a}}} \\ 0&:&{{\text{otherwise}}} \end{array}} \right.\] Which one of the following statements is true?
  20. If three coins are tossed simultaneously, the probability of getting at least one head is
  21. Consider a random variable to which a Poisson distribution is best fitted. It happens that P(x = 1) = $$\frac{2}{3}$$P(x = 2) on this distribution plot. The variance of this distribution will be
  22. A point is randomly selected with uniform probability in the X-Y. plane within the rectangle with corners at (0, 0), (1, 0), (1, 2) and (0, 2). If P is the length of the position vector of the point, the expected value of p2 is
  23. Let P(E) denote the probability of the event E. Given P(A) = 1, P(B) = $$\frac{1}{2}$$, the values of $${\text{P}}\left( {\frac{{\text{A}}}{{\text{B}}}} \right)$$ and $${\text{P}}\left( {\frac{{\text{B}}}{{\text{A}}}} \right)$$ respectively are
  24. An examination consists of two papers, Paper 1 and Paper 2. The probability of failing in Paper 1 is 0.3 and that in Paper 2 is 0.2. Given that a student has failed in Paper 2, the probability of failing in Paper 1 is 0.6. The probability of a student failing in both the papers is
  25. A lot has 10% defective items. Ten items are chosen randomly from this lot. The probability that exactly 2 of the chosen items are defective is