Statistics Question

Assignment II STATISTICS AND PROBABILITY [262 MATH-3] Academic Year 2023-24 G, 1444-45 H Third Semester DEPARTMENT OF MATHEMATICS COLLEGE OF SCIENCE, Jazan University, KSA 262 MATH-3_Third Semester_2023-2024 Name I.D. Group Number 262 MATH-3_Third Semester_2023-2024 Q 1 If 𝑋 is a random variable with distribution 𝑋: 0 1 2 3 𝑃(𝑋): 1 1 1 1 4 4 4 4 Find the mean, Variance and standard deviation. Q 2 If 𝑋 is a random variable with distribution 𝑋: 0 1 2 3 4 𝑃(𝑋): 0 𝐾 2𝐾 3𝐾 4𝐾 Explain the following: (i) For which value of 𝐾, 𝑃(𝑋) is probability mass function? (ii) Find the following probabilities: (a) 𝑃(1 ≤ 𝑋 < 4). (b) 𝑃(2 < 𝑋 ≤ 4). Q 3 A box contains 6 balls. Two are numbered 4, two are numbered 5, and two are numbered 6. The balls are mixed and one is selected at random. After a ball is selected, its number is recorded. Then it is replaced. If the experiment is repeated many times, find the variance (𝜎 2 ) and standard deviation (𝜎) of the numbers on the balls. Q 4 A special six-sided die is made in which 2 sides have 6 spots, 3 sides have 4 spots, and 1 side has 1 spot. If the die is rolled, find the expected value of the number of spots that will occur. Q 5 Let 𝑋~𝐵(4, 0.6). Find the following (i) (ii) (iii) 𝑃(𝑋 = 2) 𝑃(1 < 𝑋 < 4) Also find mean and variance. Q 6 Let 𝑋~𝑃(2). Find the following (i) (ii) (iii) 𝑃(𝑋 = 0; 2) 𝑃(𝑋 = 2; 2) Also find mean and variance. 262 MATH-3_Third Semester_2023-2024 Q 7 From the paired data: 𝑋 1 2 3 4 5 𝑌 10 8 12 12 16 (i) Find the correlation coefficient (𝑟). (ii) Explain the relationship. (iii) Find the line of regression. (iv) Estimate the value of 𝑌 when 𝑋 = 5. Q 8 Find the following probabilities or area. (i) 𝑃(𝑧 < 1.23) (ii) 𝑃(𝑧 > 1.37) (iii) 𝑃(1.45 < 𝑧 < 1.62) (iv) 𝑃(𝑧 < −1.42) (v) 𝑃(−1.34 < 𝑧 < 2.15) Q 9 From the paired data set (𝑋, 𝑌): (5,9), (4, 7), (3, 5), (2,3), (1, 1) (i) Find the correlation coefficient (𝑟). (ii) Explain the relationship. (iii) Find the line of regression. (iv) Estimate the value of 𝑌 when 𝑋 = 10. Q 10 If 𝑋 is a normal random variable with mean 4 and variance 25, then find the standard normal variate z for the following data: (i) 𝑋 = 10 (ii) 𝑋 = 3. 262 MATH-3_Third Semester_2023-2024