Let E1, E2, E, and Ebe arbitrary events. Write the following events in set notations:

i) not more than one of E₁, E₂, E₃, E₄

ii) one and only one of E₁, E₂, E₃, E₄.

iii) E₁ and at least one of E₂, E₃, E₄

iv) none of E₂, E₃, and Eusing E₁.

If X is a random variable such that E(X)=3 and E(X²) = 13, determine a lower bound for P(-2< X <8).

The mean and standard deviation of a variable x are m and  ørespectively. Obtain the mean and standard deviation of       where a,b and care constants.

Consider the joint probability density function

Are both x and y regressions linear? Give reasons for your answer.

A die is thrown 60 times with the following results:

Test that the die is unbiased at 5% level of significance. Given that at 5, 6 and 7 d.f. the value of are 11.070, 15.592 and 14.067 respectively.

6 observations on (X ,Y) yielded the following data:

i) Determine the correlation coefficient between X and Y.

ii) Given X =10, what will be the predicted value of Y ?

iii) Given Y =15,what will be the predicted value of X ?

The mean I.Q. of a large number of children of age 14 was 100 and standard deviation 16. Assuming that the distribution was normal, find

i) the percentage of children having I.Q. under 80.

ii) the limits in which the I.Q. of the middle 40% of the children will lie.

You may like to use the following values:

P(Z>1.25) = 0.1056

P(Z< -0.525) = 0.3

Which of the following statements are True or False? Give short proof or counter example in your answer.

i) If the correlation coefficient between X and Y is 0.8, then the correlation coefficient between 2X-1and-3Y-lis-0.48.

ii) If X and Y are independent binomial variates with parameters (n₁, p₁) and (n₂, p₂) respectively, then X + Y has binomial distribution with parameters (n₁+n₂, p₁ + p₂).

iii) The function defined as

is a probability density function.

iv) For a normal distribution with mean μand variance σ², the hypotheses

Η₁: μ = μο, σ² = 1 and

Η₂: μ =μο, σ² ≥1 are simple hypotheses.

v) In a problem of testing of a simple hypothesis against a simple alternative, if the probability of type-I error is known to be 0.06, then the power of the test will be 0.94.

Determine a unique polynomial f(x) of degree ≤3 such that 

Determine the spacing h in a table of equally spaced values for the function

so that the quadratic interpolation in this table satisfies | error |≤10^-6

The iteration method where N is positive constant, converges to some quantity. Determine this quantity. Also find the rate of convergence of this method.

Find the solution of the difference equation Also find the particular solution when y₀. =1 and y₁ =6. 

Use Runge-Kutta method of order four to solve y'=x+y. Start with x 1, y=0 and carry to x=1.5 with h = 0.1. 

Suppose f_n denotes the value of then find the value of

For the table of values of f (x)=xe^x given by

Find f"(2.0) using the central difference formula of 0(h²) for h=0.1 and h=0.2. Calculate T.E. and actual erroг.

Take 10 figure logarithm to base 10 from x=300 to x=310 by unit increment. Calculate the first derivative of logx when x=310. 

A solid of revolution is formed by rotating about the x-axis the area bounded between x=0, x=1 and the curve given by the table

Find the volume of the solid so formed using

i) Trapezodial rule

ii) Simpson's rule

The position f(x) of a particle moving in a line at various times x_k, is given in the following table. Estimate the velocity and acceleration of the particle at x = 1.2

Solve the  using R-K method of 0(h⁴) with h = 0.1 and obtain the value of y(0.2). Also find the error at t=0.2, if the exact solution is y(t)-t+e.

Consider the following data

Use Stirling's formula to approximate f(1.5) with x_o 1.6.