Introduction to Probability and Statistics (AL-302) - Important Questions

A random variable X has the following probability function for x = 0 to 7 with parameter K. Determine the value of K, evaluate P(X < 6) and P(0 <= X <= 4), and find the distribution function of X.

State Chebyshev's inequality and prove that the binomial distribution tends to Poisson distribution as n -> infinity, p -> 0 with np = lambda constant.

The density function of a random variable X is f(x) = e^{-x}, x >= 0. Find E(X), E(X^2) and variance of X.

Define exponential distribution with parameter theta and obtain its mean, variance and moment generating function.

Suppose a continuous random variable X has the probability density function f(x) = K(1-x^2) for 0 < x < 1, and f(x) = 0 otherwise. Find the normalizing constant K, mean and variance.

Define bivariate distribution and explain its properties.

Average number of accidents on any day on a national highway is 1.6. Determine the probability that the number of accidents are (i) at least one (ii) at most one.

Find Karl Pearson's coefficient of correlation from the following data: X: 12, 9, 8, 10, 11, 13, 7; Y: 14, 8, 6, 9, 11, 12, 3.

Given the first four central moments mu1 = 0, mu2 = 9.2, mu3 = 3.6 and mu4 = 59.4, calculate beta1, beta2, coefficient of skewness and kurtosis and comment on the nature of distribution.

Define Binomial distribution and obtain its mean and variance.

Calculate Spearman's rank correlation coefficient from the given raw X and Y data (including tied values).

Fit a second-degree parabola y = a + bx + cx^2 to the given data by the method of least squares.

Experience has shown that 20% of a manufactured product is of the top quality. In one day's production of 400 articles only 50 are of top quality. Test the hypothesis at 0.05 level using a large-sample Z-test for single proportion.

What do you understand by Chi-square test of goodness of fit? Write conditions for applying Chi-square test.

Determine (i) P(B|A) (ii) P(A|B^c) if A and B are events with P(A) = 1/3, P(B) = 1/4, P(A union B) = 1/2.