MATHEMATICS—II (Probability and Statistics)

Given $ P(A) = 0.35, P(B) = 0.63 $ and $ P(A \cap B) = 0.32 $. Find $ P(B|A) $.

For which of the following experiments is the binomial distribution an appropriate model of the experiment's probability distribution?

Consider the normal random variable X with mean $ \mu = 200 $ and standard deviation $ \sigma = 25 $. Which of the following statements is always true?

The moment-generating function of a continuous random variable X be given as $ M_X(t) = (1-t)^{-9}, |t| < 1 $. Then its mean and variance is

Which one of the following statements is always true?

The variance of first n natural number is

Kurtosis in frequency distribution is adjudged around

Which one of the following statements is always true?

Which of the following symbols is commonly used for the population variance?

Use of the chi-square statistics requires that each of the expected cell counts is

The manufacturing department of a company hires technicians who are college graduates as well as technicians who are not college graduates. Under their diversity program, the manager of any given department is careful to hire both male and female technicians. The data in table given below show a classification of all technicians in a selected department by qualification and gender. Suppose that the manager promotes one of the technicians to a supervisory position. If the promoted technician is a woman, then what is the probability that she is a non-graduate?

Classification of technicians by qualification and gender

A random variable X has the following probability function (represented here as CDF) :

(i) Find mean and variance of random variable X.

(ii) Find $ P(X = -3 | X < 0) $ and $ P(X \ge 3 | X > 0) $.

An irregular six-faced die is thrown and the expectation that 10 throws it will give five even numbers is twice the expectation that it will give four even numbers. How many times in 10000 sets of 10 throws each you would expect it to give no even number?

If a sample size n is taken from a lot of N items containing 10% defectives, show by using the Chebyshev's inequality that the probability exceeds 0.99 that the number of defectives in the sample differs from n/10 by not more than $ 3\sqrt{n} \sqrt{(N-n)/(N-1)} $.

Find the mean of normal distribution.

Suppose that the lapse of time between two successive accidents in a paper mill is exponentially distributed with a mean of 15 days. Find the probability that the time between two successive accidents at that mill is more than 20 days.

Three coins are tossed. Let X denote the number of heads on the first two coins, Y denote the number of tails on the last two and Z denote the number of heads on the last two.

(a) Find the joint distribution of (i) X and Y, (ii) X and Z.

(b) Find the conditional distribution of Y given X = 1.

(c) Find $ E(Z | X = 1) $.

(d) Find $ \rho_{X, Y} $ and $ \rho_{X, Z} $.

(e) Give a joint distribution that is not the joint distribution of X and Z in part (a) and yet has the same marginals as $ f(x, z) $ has in part (a).

Find the first four moments (i) about the origin and (ii) about the mean for a random variable X having density function

$ f(x) = \begin{cases} 4x(9 - x^2)/81, & 0 \le x \le 3 \\ 0, & \text{otherwise} \end{cases} $

The number of defective parts produced per shift can be modeled using a random variable that has the Poisson distribution. Assume that, on average, three defective parts per shift are produced.

(i) What is the probability that exactly four defective parts are produced in a given shift?

(ii) What is the probability that more than seven defective parts are produced in the next two shifts?

A manufacturer knows from experience that the diameters of 0.250 in. precision-made pins he produces have a normal distribution with mean 0.25000 in. and standard deviation 0.00025 in. What percentages of the pins have diameters between 0.24951 in. and 0.25049 in? This question is equivalent to find the probability that the diameter, say X, of a pin taken at random from the production lies between 0.24951 in. and 0.25049 in.

Fit a straight line to the following data :

The variables X and Y are connected by the equation $ aX + bY + c = 0 $. Show that the correlation between them is -1, if the signs of a and b are alike and +1, if the signs of a and b are different.

A random sample of 500 apples was taken from a large consignment and 60 were found to be bad. Obtain the 98% confidence limits for the percentage of bad apples in the consignment.

The means of two single-large samples of 1000 and 2000 members are 67.5 inches and 68.0 inches respectively. Can the samples be regarded as drawn from the same population of standard deviation 2.5 inches? (Test at 5% level of significance.)

Describe the chi-squared test for testing a hypothesis that a normal population has a specified variance $ \sigma^2 $.

If $ A $ and $ B $ are independent, then show that $ \bar{A} $ and $ B $ are also independent.

Let $ A $ and $ B $ are two possible outcomes of an experiment and suppose $ P(A)=0.4 $, $ P(A \cup B)=0.7 $ and $ P(B)=k $.

(i) For what choice of 'k' $ A $ and $ B $ are mutually exclusive?

(ii) For what choice of 'k' $ A $ and $ B $ are independent?

If $ n $ people are seated at a round table, what is the chance that two named individuals will be next to each other?

A five figure number is formed by digits 0, 1, 2, 3, 4 (without repetition). Find the probability that the number formed is divisible by 4.

A point is chosen at random out of four points in 3-dimensional space (1, 0, 0), (0, 1, 0), (0, 0, 1), (1, 1, 1). Let $ E_i = i $-th coordinate is 1. Check if the events $ E_1, E_2 $ and $ E_3 $ are independent.

The members of a consulting firm rent cars from three rental agencies; 60% from agency A, 30% from agency B and remaining from agency C. If 9% of the cars from agency A need a tune-up, 20% from B, and 6% of the cars from C need a tune-up, what is the probability that a rental car needs a tune-up, when it came from agency B?

If $ X $ and $ Y $ are two independent random variables with $ E(X)=\alpha $, $ E(X^2)=\beta $ and $ E(Y^k)=a_k $, $ k=1, 2, 3, 4 $. Find $ E(XY+Y^2)^2 $.

A discrete random variable $ X $ has probability mass function $ f(x) = k \left( \frac{1}{2} \right)^x $, $ x = 1, 2, 3, 4, \dots $. Find the value of $ k $.

The incidence of occupational disease in an industry is such that workers have a 25% chance of suffering from it. What is the probability that out of 13 workers chosen at random, 5 or more will suffer from the disease?

A random variable $ X $ follows the binomial distribution with $ B(40, 1/4) $. Use Chebyshev's inequality to find bound for $ P(|X-10|>10) $.

The following marks have been obtained by a class of students in Statistics (out of 100):

Paper-I : 80, 45, 55, 56, 58, 60, 65, 68, 70, 75, 85

Paper-II : 82, 56, 50, 48, 60, 62, 64, 65, 70, 74, 90

Compute the coefficient of correlation for the above data.

Fit a second degree parabola to the following data, $ x $ is the independent variable :

A language class has only three students A, B, C and they independently attend the classes. The probability of attendance of A, B and C on any given day is $ 1/2, 2/3 $ and $ 3/4 $ respectively. Find the probability that total number of attendance in two consecutive days is exactly 3.

A bag contains $ n $ white and 2 black balls. Balls are drawn at random one by one without replacement until a black ball is drawn. If $ K $ white balls are drawn before first black ball, a man is to receive $ K^2 $. Find his expected gain.

The lines of regression for a bivariate population are $ Y=X $ and $ 4X-Y=3 $, and that the second moment of $ X $ about the origin is 2. Find (i) the correlation coefficient (r) and (ii) the standard deviation of $ Y $.

Ball bearings of a given brand weighs 15 gram with a standard deviation of 0.5 gram. What is the probability that two lots of 1000 ball bearings each will differ in weight by more than 50 gram?

$ P[ (A \cup B) / C ] = P(A/C) + P(B/C) - P[ (A \cap B) / C ] $

$ P[ (A \cap \bar{B}) / C ] = P(A/C) - P[ (A \cap B) / C ] $

Check whether this is a probability density function.

Find the mean of $ X $.

Find the variance of $ X $.

Find $ a $ and $ b $ such that $ P(X \le a) = P(X > a) $ and $ P(X > b) = 0.05 $.

Find $ P(0.2 < X < 0.5) $.

Find $ P(X < 0.3) $.

Find $ P(X > 0.75 / X > 0.50) $.

Find $ C $;

Find $ P(X < 1/2, Y < 1/2) $;

Find $ P(1/4 < X < 3/4) $;

Find $ P(|X| > 1) $.

A random variable follows Gamma ($ \alpha, \beta $) distribution with probability density function $ f_X(x) = \begin{cases} \frac{1}{\Gamma(\alpha)\beta^\alpha} x^{\alpha-1} e^{-x/\beta}, & x \ge 0 \\ 0, & \text{elsewhere} \end{cases} $. If $ X $ and $ Y $ are independent gamma ($ \alpha_1, \beta $) and gamma ($ \alpha_2, \beta $), then find the probability density function $ X+Y $.

It has been found from experience that the mean breaking strength of a particular brand of thread is 275.6 gram with a standard deviation of 39.7 gram. Recently a sample of 36 pieces of thread showed a mean breaking strength of 253.2 gram. Can one conclude at a significance level (i) 0.05 and (ii) 0.01 level that the thread has become inferior?

On an elementary school examination in spelling, the mean grade of 32 boys was 72 with a standard deviation of 8, while the mean grade of 36 girls was 75 with a standard deviation of 6. Test the hypothesis at a (i) 0.05 and (ii) 0.01 level of significance that the girls are better in spelling than the boys.