Test: Probability and Statistics - 6 - Mathematics MCQ

In a tank containing three fish A, B, C, pellets of food are frequently placed. Each time a pellet is dropped, the fish compete for it. Suppose that over a long period of time it is observed that either A or B is successful 1/2 of the time and that either A or C is successful 3/4 of the time. What is the probability that A is successful?

If four numbers are selected at random from the ten numbers 1, 2, 3, .........10, repetitions being allowed, then the probability that the product of the numbers is of the form 3n - 1 or 3n + 1, n being a non-negative integer, is

A fair die is tossed repeatedly until 6 shows up 3 times. The probability’ that exactly 5 tosses are needed is

The probability that a person tossing three fair- coins will get either all heads or fails for the second time on the 5^th toss is

There are two groups of subjects, one of which consists of 5 science and 3 engineering subjects and the other consists of 3 science and 5 engineering subjects, an unbiased die is cast. If the number 3 or 5 turns up a subject is selected at random from the first group. What is the probability that an engineering subject is ultimately selected?

A coin is tossed until a heard appears or until the coin has been tossed 5 times. If a head does not occur on the first two losses, then the probability that the coin will be tossed 5 times is

M telegrams are distributed at random over N communication channels (V > M). The probability of the event
A = {not more than one telegram will be sent over any channel} is

Let (n, p) and λ be the parameters of binomial and poisson distributions respectively. Consider the statements
P. The mean of the binomial distribution is np
Q. The standard deviation of the binomial distribution is np (1 - p)
R. The mean of the poisson distribution is λ
S The variance of the poisson distribution is λ
Which of the following group of statements is correct?

Let X and Y be independent Poisson random variables with parameters 5 and μ, respectively. If P(XY= 1) = P(X+ Y= 1), then Var(Y) is

Let X and V be binomial random variables with parameters
(n₁, p₁) and {n₂, p₂) respectively, where 0 < p_1^, p₂ < 1, p₁ ≠ p₂. consider the following statements : 
P : If E(X)= Var(Y) then Var(X) < E(Y)
Q : If E(X) = E(Y) then Var (X) = Var(Y)
Which one of the following is TRUE?

Let X be a binomial random variable with parameters (5, p). The values of p for which P( | X - E(X) | ≤ 3= 1 are  given by

Let I_A and I_B be indicator variables for the events A and B such that 

The covariance of I_A and I_B is 

 equals to

If X₁ arid X₂ are independent binomial variates with parameters n₁ = 3, p₁ = 1 /3 and n₂ = 5. p₂  = 1/3, then P(X₁ + X₂ ≥ 1) is

Consider a normalized floating point number in base β so that mantissa X satisfies the condition (1/β) ≤ X < 1. Experience shows that X has the following probability density function f_x(x) = k / x , k > 0. The value of k is

Let x₁, x₂, ..., x_n be a random sample drawn from normal population with mean μ and variance σ². Writing  the statistics  follows

X is a random variable with p.d.f. f(x) =1 /2a, -a < x < a. E(e^tX) equals to