Q.1) A random variable X has the following probability function:

Q.2) Define discrete random variable and independence of random variable. Also, show that $E(X_{1}+X_{2}+...+X_{n})=E(X_{1})+E(X_{2})+...+E(X_{n})$. (Dec-2023)

Define discrete random variables. If X and Y are any two random variables, then show that $E(X+Y) = E(X) + E(Y)$ Provided $E(X)$ and $E(Y)$ exist. (June-2024)

Q.3) Define expectation of random variables. Also, show that $V(aX+b)=a^{2}V(X)$. (Dec-2023)

Define non-central and central moments. Also, show that $V(2X+3)=4V(X)$. (June-2023)

If X is a continuous random variable and $k$ is a constant. Then prove that $Var(X+k)=Var(X)$ and $Var(kX)=k^{2}Var(X)$. (Dec-2024)

Q.4) State Chebyshev's inequality. If $n \rightarrow \infty,$ $p \rightarrow 0$ and $np=\lambda$ then show that binomial distribution reduces to Poisson distribution. (Dec-2023)

Show that Poisson distribution is a limiting case of binomial distribution under the case $n \rightarrow \infty$, $p \rightarrow 0$ and $np=m$. (June-2023)

State and Prove Chebyshev's inequality. (Dec-2024)

Define: Chebyshev's inequality. (Nov-2022)

Q.5) A sample of 4 items is selected at random from a box containing 12 items of which 5 are defective. Find the expected number E of defective items. (June-2024)

Q.6) A random variable X is defined as the sum of the numbers on the faces when two dice are thrown. Find the mean of X. (Dec-2024)

Q.7) Show that sum of two independent random variables follows normal distribution. (June-2023)

Q.8) Determine (i) $P(B/A)$ (ii) $P(A/B^c)$ if A and B are events with $P(A) = \frac{1}{3}, P(B) = \frac{1}{4}, P(A \cup B) = \frac{1}{2}$ (June-2025)

Q.9) A random variable X has the following probability function.

Find i) K ii) Mean iii) Variance (June-2025)

Q.10) Define Moments. Explain types of moments. (June-2025)

Q.1) What is a multinomial distribution? Find its mean and variance. (Predicted)

Q.2) State the axioms of probability. Prove that $P(A \cup B) = P(A) + P(B) - P(A \cap B)$. (Predicted)

Q.3) Explain the concept of Infinite sequences of Bernoulli trials. (Predicted)

Q.4) Differentiate between Moment Generating Function (MGF) and Characteristic Function. (Predicted)

Q.5) If X is a random variable with mean $\mu$ and variance $\sigma^2$, prove that for any $k > 0$, $P(|X-\mu| \ge k\sigma) \le \frac{1}{k^2}$. (Predicted)

Q.1) The density function of a random variable X is $f(x)=e^{-x}$ when $x \ge 0$. Find $E(x)$, $E(x^{2})$ and variance of X. (Nov-2022)

Q.2) 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: i) K ii) Mean iii) Variance. (Nov-2022)

Q.3) Define gamma distribution. If X follows exponential distribution with parameter $\theta$, then obtain its mean and variance. (Dec-2023, June-2023)

Define gamma distribution with parameter $\lambda$ and obtain its mean, variance, and characteristic function. (June-2023)

Explain Exponential and Gamma densities with properties. (Dec-2024)

Define exponential distribution with parameter $\lambda$ and obtain its mean, variance and moment generating function. (Dec-2023)

Q.4) If $X \sim N(\mu, \sigma^2)$, then show that $\varphi_{x}(t) = e^{i\mu t - \frac{1}{2}t^2\sigma^2}$. (June-2023)

Q.5) A large number of measurement is normally distributed with a mean 65.5 cm and S.D. of 6.2 cm. Find the percentage of measurement that fall between 54.8 cm and 68.8 cm. (June-2023)

The mean and standard deviation of a normal variable are 8 and 4 respectively. Find: $P(5\le X\le10)$ and $P(X\ge5)$. (Dec-2024)

If X is normally distributed with mean 2 and variance 0.1, then find $P(|x-2| \ge 0.01)$. (June-2024)

Q.6) Define continuous random variable. Also show that $E(X_{1}X_{2}...X_{n}) = E(X_{1})E(X_{2})...E(X_{n})$. (June-2023)

Q.7) The probability density function of a random variable X is $f(x) = \frac{\sin x}{2}$ for $0 \le x \le \pi$ and $0$ elsewhere. Find the mean, mode and median of a distribution and also find the probability between 0 and $\frac{\pi}{2}$. (Dec-2024)

Q.8) If X is a continuous random variable and $Y = aX + b$. Prove that $E(Y) = aE(X) + b$ and $V(Y) = a^2V(X)$. (June-2024)

Q.9) The probability density $f(X)$ of a continuous random variable is given by $f(x) = Ce^{-|x|}, -\infty < x < \infty$. Show that $C=\frac{1}{2}$ and find the mean and Variance of the distribution. Also find the probability that the variate lies between 0 and 4. (June-2024)

Q.10) Let $f(X)$ be the probability density function of a continuous random variable X. Then explain Mean, Median, Mode, variance and mean deviation. (June-2024)

Q.11) Find the mean and standard deviation of a normal distribution in which 7% of items are under 35 and 89% are under 63. (June-2025)

Q.12) The density function of a random variable X is $f(x) = e^{-x}$, $x \ge 0$. Find $E(x)$, $E(x^2)$ and variance of X. (June-2025)

Q.13) Define gamma distribution. Find the mean and Variance of gamma distribution. (June-2025)

Q.1) Obtain the Moment Generating Function (MGF) of the Gamma Distribution. (Predicted)

Q.2) Discuss the memory-less property of Exponential Distribution. (Predicted)

Q.3) Find the Cumulative Distribution Function (CDF) for the standard normal distribution. (Predicted)

Q.4) Explain the relationship between Gamma and Chi-square distributions. (Predicted)

Q.5) Verify that the total area under the normal curve is 1. (Predicted)

Q.1) Define Bivariate distribution. Explain their properties. / Describe bivariate distribution. (Nov-2022, Dec-2023, Dec-2024)

Write down the probability density function of bivariate normal distribution. (Dec-2023)

Q.2) Define Baye's theorem. (Nov-2022)

Q.3) The chance that doctor A will diagnose a disease X correctly is 60%. The chance that a patient will die by his treatment after correct diagnosis is 40% and the chance of death by wrong diagnosis is 70%. A patient of doctor A, who had disease X, died. What is the chance that his disease was diagnosed correctly. (Nov-2022)

Q.4) State and prove Bayes theorem. Suppose 5 men out of 100 and 25 Women out of 10,000 are color blind. A color blind person is chosen at random. What is the probability of the person being a male. (Assuming male and female to be in equal numbers). (June-2024)

Q.5) If first box contains 2 black, 3 red, 1 white balls; Second box contains 1 black, 1 red, 2 white balls; Third box contains 5 black, 3 red, 4 white balls. Of these a box is selected at random. From it a red ball is randomly drawn. If the ball is red, find the probability that it is from second box. (Dec-2024)

Q.6) Two factories produce identical clocks. The production of the first factory consists of 10,000 clocks of which 100 are defective. The second factory produces 20,000 clocks of which 300 are defective. What is the probability that a particular defective clock was produced in the first factory. (June-2025)

Q.7) Explain bivariate distributions and their properties. (June-2025)

Q.1) What are Marginal and Conditional distributions in the context of Bivariate Random Variables? (Predicted)

Q.2) Explain the independence of random variables in a Bivariate distribution. (Predicted)

Q.3) Discuss the distribution of sums and quotients of random variables. (Predicted)

Q.4) Find the conditional density of X given Y=y for the joint PDF $f(x,y) = x+y$ for $0(Predicted)

Q.5) Explain the concept of Correlation coefficient in terms of Bivariate expectations $E(XY)$. (Predicted)

Q.1) The mean and variance of a binomial variable X with parameters n and P are 16 and 8 respectively. Find $P(X \ge 1)$ and $P(X > 2)$. (Nov-2022, June-2024)

Q.2) A manufacturer knows that the condensers he makes contain on average 1% defectives. He packs them in boxes of 100. What is the probability that a box picked at random will contain 3 or more faulty condensers? (Nov-2022)

Q.3) If X is a normal variate with mean 30 and standard deviation is 5. Find i) $P(26 \le X \le 40)$ ii) $P(X \ge 45)$. (Nov-2022)

Q.4) Find Karl pearson's coefficient of correlation from the following data: (Nov-2022)

Calculate coefficient of correlation from the following data. (Dec-2024)

Define Karl Pearson's correlation coefficient and obtain the correlation coefficient for the given data (refer to June-2023 paper for data table). (June-2023)

Q.5) Define regression coefficient with its properties. If $X \sim N(\mu,\sigma^{2})$, then show that $M_{x}(t)=e^{\mu t+\frac{1}{2}t^{2}\sigma^{2}}$. (Dec-2023)

Q.6) What do you understand by measure of central tendency? Also, write down its merits and demerits. Calculate mean and standard deviation for the observations 5, 10, 20, 25, 40, 42, 45, 48, 70, 80. (Dec-2023)

Write down merits and demerits of measure of central tendency. Find the median for the following data (Wages vs No of workers table). (June-2023)

Q.7) Define Binomial distribution and obtain its mean and variance. (Dec-2023)

Define binomial distribution and derive its moment generating function. Hence, obtain its mean and variance. (June-2023)

Q.8) Define Poisson distribution and obtain its mean and variance. (June-2023)

Q.9) What do you mean by measure of skewness? Write tests of skewness. (Dec-2023)

Q.10) Define Spearman's rank correlation coefficient and obtain rank correlation coefficient for the given data (refer to Dec-2023 paper). (Dec-2023)

Find the rank correlation coefficient to the following data (refer to June-2023 paper). (June-2023)

Q.11) What do you mean by measure of kurtosis? Obtain $\beta_{1}$ and $\beta_{2}$ for the function: $f(x) = y_{0}x(2-x)$; $0 \le x \le 2$. (June-2023)

The first four central moments of a distribution are 0, 2.5, 0.7 and 1.75. Calculate $\beta_1, \beta_2$. (June-2024)

The values of $\mu_{1},$ $\mu_{2}$, $\mu_{3}$ and $\mu_{4}$ are 0, 9.2, 3.6 and 1.22 respectively. Find out the Skewness and Kurtosis of the distribution. (Dec-2024)

Q.12) Find the regression line of y on x for the following data (refer to Dec-2023 paper). (Dec-2023)

Q.13) Explain comparison between Correlation and Regression. (June-2024)

Q.14) Define correlation. Explain types of correlation and methods of studying correlation. (Dec-2024)

Define: Correlation coefficient. (Nov-2022)

Q.15) 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. (Dec-2024)

Q.16) If the variance of a Poisson variate is 3, then find $P(x=0)$, $P(0 < x \le 3)$ and $P(1 \le x < 4)$. (June-2025)

Q.17) 20% of items produced from a factory are defective. Find the probability that in a sample of 5 chosen at random. i) none is defective ii) one is defective iii) $P(1 < x < 4)$. (June-2025)

Q.18) Calculate the coefficient of rank correlation from the below data. (June-2025)

Q.19) Explain partial and multiple correlation coefficients. (June-2025)

Q.1) Show that the correlation coefficient lies between -1 and +1. (Predicted)

Q.2) Derive the regression lines X on Y and Y on X from the method of least squares. (Predicted)

Q.3) What is the relationship between Central Moments and Moments about origin? Derive the first 4 relations. (Predicted)

Q.4) Explain properties of Normal Curve. Why is it called a symmetric distribution? (Predicted)

Q.5) If X and Y are independent variables, show that the coefficient of correlation between them is zero. Is the converse true? (Predicted)

Q.1) By the method of least squares fit a parabola of the form $y=a+bx+cx^{2}$ for the following data. (Nov-2022)

Q.2) It is claimed that a random sample of 49 tyres has mean life of 15200 km. This sampled was drawn from a population whose mean is 15150 km's and a standard deviation of 1200 km. Test the significance at 0.05 level. (Nov-2022)

Q.3) Describe the methodology for difference of means for large samples. (Dec-2023)

Q.4) Fit a second degree parabola to the following data (X: 10 to 40, Y: 11 to 41). (Dec-2023)

Q.5) Using method of Least squares, find the curve $y=ax+ax^{2}$ that best fit the following data (X: 1 to 5, Y: 1.8 to 19.8). (June-2023)

Q.6) In a big city 325 men out of 600 men were found to be smokers. Does this information support the conclusion that the majority of men in this city are smokers. (June-2023)

Q.7) Fit a Parabola for the following data (X: 0 to 4, Y: 1 to 6.3). (June-2024)

Q.8) Fit a Straight line for the following data. (Dec-2024)

Q.9) A sample of 400 items is taken from a population whose standard deviation is 10. The mean of the sample is 40. Test whether the sample has come from a population with mean 38. Also calculate 95% confidence interval for the population. (Dec-2024)

Q.10) Fit a straight line to the form $y = a + bx$ for the following data. (June-2025)

Q.11) The average marks scored by 32 boys in 72 with a standard deviation of 8. While that for 36 girls is 70 with a standard deviation of 6, does this indicate that the boys perform better than girls at level of significance 0.05? Explain. (June-2025)

Q.12) Experience had show 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. (June-2025)

Q.1) Explain the procedure for Testing of Hypothesis. Define Null Hypothesis, Alternative Hypothesis, and Level of Significance. (Predicted)

Q.2) Fit an exponential curve $y = ae^{bx}$ to the given data. (Predicted)

Q.3) A machine produces 16 defective articles in a batch of 500. After overhauling, it produces 3 defectives in a batch of 100. Has the machine improved? (Test for difference of proportions). (Predicted)

Q.4) Differentiate between One-tailed and Two-tailed tests with examples. (Predicted)

Q.5) Explain Type I and Type II errors in testing of hypothesis. (Predicted)

Q.1) A sample of 26 bulbs gives a mean life of 900 hours with a standard deviation of 20 hours. The manufacturer claims that the mean life of bulbs is 1000 hours. Is the sample not up to the standard. (5% L.O.S.) (Nov-2022)

Q.2) In one sample of 10 observations, the sum of the squares of the deviations of the sample values from mean was 120 and in the other sample of 12 observations, it was 314. Test whether the difference is significant at 5% level? (Nov-2022)

Q.3) What do you understand by Chi-square test of goodness of fit? Write condition for applying Chi-square test. (Dec-2023)

Define Chi-Square test of goodness of fit. Explain the conditions for Chi-Square test. (Dec-2024)

Q.4) Out of 8000 graduates in a town, 800 are females, out of 1600 graduate employees, 120 are females, Use $\chi^2$ test to determine if any distinction is made in appointment on the basis of sex. The value of $\chi^2$ for 1 degree of freedom at 5% level is 3.841. (June-2023)

Q.5) The table below give the number of air craft accidents that occurred during the various days of the week. Test whether the accidents are uniformly distributed over the week. (Dec-2024)

Q.6) Define chi-square distribution. Explain it's properties. (June-2025)

Q.1) What is t-distribution? Explain the properties and applications of t-distribution. (Predicted)

Q.2) Two independent samples of 8 and 7 items respectively had the following values:
Sample 1: 9, 11, 13, 11, 15, 9, 12, 14
Sample 2: 10, 12, 10, 14, 9, 8, 10
Is the difference between the means of the two samples significant? (Predicted)

Q.3) Explain F-test for equality of two population variances. (Predicted)

Q.4) What are degrees of freedom? How are they calculated for t-test and Chi-square test? (Predicted)

Q.5) Discuss the Student's t-test for single mean. What are the assumptions made? (Predicted)