Mathematics

Probability Distributions

488 Questions

Probability distributions describe the likelihood of different outcomes in a statistical experiment. Key topics include calculating confidence intervals, understanding Type II errors, and working with the standard normal distribution. These concepts are frequently tested in mathematics and statistics sections of various competitive exams.

Standard normal distributionType II error probabilityConfidence interval interpretationPoisson distributionProbability density function

Probability Distributions Questions

Multiple choice

X(t) is a random process with a constant mean value of 2 and the autocorrelation function Rx ($\tau$) = 4$\lfloor e^{-0.2 | d} + 1 \rfloor$.

Let X be the Gaussian random variable obtained by sampling the process at t = ti and let Q ($\alpha$) = $\int_\infty^\infty \dfrac{1}{\sqrt {2\pi}} e^{\dfrac{-y^2}{2}} dy$ The probability that [ x $\le$ 1 ] is

  1. 1 - Q(0.5)

  2. Q(0.5)
  3. Q $\dfrac{1}{2 \sqrt 2}$
  4. 1-Q $\left( \dfrac{1}{2 \sqrt 2} \right)$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Multiple choice

Consider a binary digital communication system with equally likely 0s and 1s. When binary 0 is transmitted the detector input can lie between the levels - 0.25 V and + 0.25 V with equal probability: when binary 1 is transmitted, the voltage at the detector can have any value between 0 and 1 V with equal probability. If the detector has a threshold of 0.2 V (i.e., if the received signal is greater than 0.2 V, the bit is taken as 1), the average bit error probability is

  1. 0.15

  2. 0.2

  3. 0.05

  4. 0.5
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Multiple choice

Two independent random variables X and Y are uniformly distributed in the interval [-1, 1]. The probability that max [X, Y] is less than 1/2 is

  1. ¾

  2. 9/16

  3. ¼

  4. 2/3
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

X and Y are uniform on [-1,1], so PDF is f(x) = 1/2 on this interval. We need P(max(X,Y) < 1/2) = P(X < 1/2 AND Y < 1/2). Since independent: P(X < 1/2) = (1/2 - (-1)) / 2 = 1.5/2 = 3/4. Same for Y. P(max < 1/2) = P(X < 1/2) × P(Y < 1/2) = (3/4) × (3/4) = 9/16. The interval from -1 to 1/2 has length 1.5, and the total interval length is 2, so probability is 3/4 for each variable.

Multiple choice

A discrete random variable X takes values from 1 to 5 with probabilities as shown in the table. A student calculates the mean X as 3.5 and her teacher calculates the variance of X as 1.5. Which of the following statements is true?

  1. Both the student and the teacher are right.

  2. Both the student and the teacher are wrong.

  3. The student is wrong but the teacher is right.

  4. The student is right but the teacher is wrong.
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation