Define and give the example of Random variable.
[3 marks]What is the probability that a leap year selected at random will have 53 Sundays?
[4 marks]In a bolt factory, three machines A, Band Cproduce 25%, 35% and 40% of total output respectively. It was found that 5%,4% and 2% are defective bolts in the production by machines A, B, Crespectively. Abolt is chosen at random from the total output and is found to be defective. Find the probability that it is manufactured from (i) Machine A(ii) Machine B (iii) Machine C.
[7 marks]Abag contains 3 red and 4 white balls. Two draws are made without replacement. What is the probability that both balls are red.
[3 marks]The probability that a student Asolves a mathematics problem is 2 and the5 probability that a student Bsolves it is . What is the probability that (i) the3 problem is not solved (ii) both Aand Bcan solve the problem, working independently of each other?
[4 marks]Verify that the following function F(x) is a distribution function. 0; x < 0 F(x) = { x − 1−e 4; x ≥ 0 Also, find the probabilities P(X ≤ 4),P(X ≥ 8), P(4 ≤ X ≤ 8).
[7 marks]The probability mass function of a random variable Xis zero except at the points X = 0,1,2. At these points, it has the values P(X = 0) = 3c3, P(X = 1) = 4c −10c2,P(X = 2) = 5c −1. Find (i) c (ii) P(X < 1) (iii) P(1 < X ≤ 2) (iv) P(0 < X ≤ 2).
[7 marks]Find the constant k such that the function kx2; 0 < x < f(x) = { 0; otℎewise is a probability density function,
[3 marks]Arandom variable Xhas the following distribution: X 1 2 3 4 5 P(X 1 3 5 7 9 11 = x) 36 36 36 36 36 36 Find (i) mean (ii) variance.
[6 marks]Compute Karl Pearson’s coefficient of correlation between Xand Yfor the following data: x 10 14 18 22 26 30 y 18 12 24 6 30 36 OR1
[7 marks]Find coefficient of correlation between x and y if the regression lines are: x+ 6y = 6 and 3x +2y = 10.
[3 marks]Calculate the first four moments from the following data: x 0 1 2 3 4 5 6 7 y 5 10 15 20 25 20 15 10
[5 marks]The following data give the experience of machine operators and their performance rating as given by the number of good parts turned out per 100 piece. Operator 1 2 3 4 5 Performance 23 43 53 63 73 83 rating (x) Experience 5 6 7 8 9
[10 marks]Calculate the regression line of performance rating on experience and also estimate the probable performance if an operator has 11 years of experience.
Explain the term related to testing of hypothesis: (i) Null hypothesis (ii) Alternate hypothesis and (iii) Level of Significance
[3 marks]Adice is tossed 960 times and it falls with 5 upwards 184 times. Is the dice unbiased at a level of significance of 0.01? (|Z | = 2.58) 0.01
[4 marks]Aset of five similar coins is tossed 320 times and result is obtained as follows No of 0 1 2 3 4 heads Frequency 6 27 72 112 71 32 Test the hypothesis that the data follow a binomial distribution. (Critical value 𝜒2 = 11.07) 0.05
[5 marks]The heights of 10 males of a given locality are found to be 175, 168, 155, 170, 152, 170,175,160,160 and 165 cm. Based on this sample, find the 95% confidence limits for the heights of males in that locality. [t (v = 9) = 2.262] 0.05
[3 marks]Random samples drawn from two countries gave the following data relating to the heights of adult males: Country A Country B Standard 2.58 2.50 deviation (in inches) Number in 1000 1200 samples Is the difference between the standard deviation significant? (|Z | = 1.96) 0.05
[4 marks]If a random variable has a Poisson distribution such that P(X = 1) = P(X = 2), find (i) the mean of the distribution (ii) P(X = 4) (iii) P(X ≥ 1) (iv) P(1 < X < 4).
Fit a straight liney = ax +b to the following data: x 1 2 3 4 6 y 2.4 3 3.6 4 5
[6 marks]Fit a curve y = abx to the following data: x 1 2 3 4 5 6 7 y 1 1.2 1.8 2.5 3.6 4.7 6.6 9.1
[8 marks]The lifetime of a certain kind of batteries has a mean life of 400 hours and the standard deviation as 45 hours. Assuming the distribution of lifetime to be normal, find (i) the percentage of batteries with a lifetime of at least 470 hours (ii) the proportion of batteries with a lifetime between 385 and 415 hours, and (iii) the minimum life of the best 5% of batteries. (P(0 < z < 0.33) = 0.1293)
[7 marks]The mean and variance of a binomial variate are 8 and 6. Find P(X ≥ 2).
[3 marks]Out of 800 families with 5 children each, how many would you except to have (i) 3 boys? (ii) 5 girls?
[4 marks]Fit a second-degree parabolic curve to the following data: x 1 2 3 4 5 6 7 8 9 y 2 6 7 8 10 11 11 10 9
[7 marks]