List of practice Questions

Consider a probability distribution given by the density function \( P(x) \). \[ P(x) = \begin{cases} Cx^2, & \text{for } 1 \leq x \leq 4, \\ 0, & \text{for } x < 1 \text{ or } x > 4. \end{cases} \] The probability that \( x \) lies between 2 and 3, i.e., \( P(2 \leq x \leq 3) \), is ___________. (rounded off to three decimal places)
A quadratic polynomial \( (x - \alpha)(x - \beta) \) over complex numbers is said to be square invariant if \[ (x - \alpha)(x - \beta) = (x - \alpha^2)(x - \beta^2). \] Suppose from the set of all square invariant quadratic polynomials we choose one at random. The probability that the roots of the chosen polynomial are equal is ___________. (rounded off to one decimal place)
A box contains 5 coins: 4 regular coins and 1 fake coin. When a regular coin is tossed, the probability \( P({head}) = 0.5 \), and for a fake coin, \( P({head}) = 1 \). You pick a coin at random and toss it twice, and get two heads. The probability that the coin you have chosen is the fake coin is __________.
Suppose a 5-bit message is transmitted from a source to a destination through a noisy channel. The probability that a bit of the message gets flipped during transmission is 0.01. Flipping of each bit is independent of one another. The probability that the message is delivered error-free to the destination is ___________. (rounded off to three decimal places)