Probability - Statistics Assignment Help

Assignment Task

1. If p(x) = cx² for (for x=0, 1, and 2), find the value of c. Find mean and variance of the distribution.

2. For the following distribution, find mean and variance and Comment on the skewness of the above distribution.

3. When we roll a pair of balanced dice, what are the probabilities of getting (a) 8 (b) 10 (c) 8 or 10 (d) 5(e) 4 or 11 (f) 4, 5 or 11.

4. Find the mean and variance of a binomial distribution with p= ¼ and n=4.

5. The supervisor of a group of 20 construction workers wants to get the opinion of 2 of them (to be selected at random) about certain new safety regulations. If 12 workers favour the new regulations and the other 8 are against them, what is the probability that both of the workers chosen by the supervisor will be against the new safety regulations?

6. For the following random variable, find E (x ²)

Find Mean, Median, Mode Variance, and Coefficient of variance associated with this.

7. For the following random variable, find E(x), E (X²) and hence Variance of X

8. Find the mean of the probability distribution of the number of heads obtained in 4 flips of a balanced coin.

9. Consider a binomial distribution with n=30, p=0.01. Find the probabilities of x=5 and x=10. Now use Poisson distribution to compute these probabilities. Comment on the results.

10. Assume a binomial distribution with parameters (n=100, p=0.04). Tabulate the probabilities for X=0, 1, 2, etc. (up to 15). Now use Poisson distribution and compute these probabilities (wherein ? = n.p) and find the difference between them as shown in following table. Comment on these results

11. The distribution of the outcome of a throw of a dice is a discrete distribution between 1 and 6. Find: E(x), Var (X) for this distribution. Find the mean & median for the same.

12. A lot of 100 items contain 20 defectives. From this lot, a sample of 10 items is considered without replacement. For this sample, tabulate values for probabilities of getting 0, 1, 2.. up to 10 defectives. State all your assumptions.

13. Suppose flaws (cracks, chips, specks, etc.) occur on the surface of glass with density of 3 per square metre. What is the probability of there being exactly 4 flaws on a sheet of glass of area 0.5 square metre? State all assumptions here.

14. The mean IQ score for 1500 students is 100, with a standard deviation of 15. Assuming the scores have a normal curve, determine the following: a. How many have an IQ over 90? b. How many have an IQ between 90 and 120? c. How many have an IQ lower than 75?

15. For certain workers, the mean wage is Rs 55/hr, with a standard deviation of Rs 5. If a worker is chosen at random, what is the probability that the worker's wage is between Rs 47 and Rs 60?

16. A tyre company tested a particular model of tire and found the tyres to be normally distributed with respect to wear. The mean was 26,000 miles and the standard deviation was 2200 miles. If 2000 tyres are tested, about how many are likely to wear out before 23,000 miles?

17.(a) If the mean is 63 and a score of 53 corresponds to a z-vale of -1.25, what is the standard deviation? (b) The mean for a normal distribution is 50 and the standard deviation is 10. What percentile is the value of 38? What z score correspond to the 67th percentile? (c) In a normal distribution 8 % items are under 50 and 10 % are over 60. Find the mean and standard deviation of the distribution.

18. Two companies tested their products to determine the average (mean) life of their products. Company A had an average life of 150 hr with standard deviation of 10 hr. Company B had an average life of 145 hr with a standard deviation of 2 hr. Which company would you rather buy from? Explain why.

19. The probability that a person has a certain disease is 0.03. Medical diagnostic tests are available to determine whether the person actually has the disease. If the disease is actually present, the probability that the medical diagnostic test will give a positive result (indicating that the disease is present) is 0.90. If the disease is not actually present, the probability of a positive test result (indicating that the disease is present) is 0.02. Suppose that the medical diagnostic test has given a positive result (indicating that the disease is present). What is the probability that the disease is actually present? What is the probability of a positive test result?

20. Customers arrive at the drive-up window of a fast-food restaurant at a rate of 2 per minute during the lunch hour. a. What is the probability that the next customer will arrive within 1 minute? b. What is the probability that the next customer will arrive within 5 minutes? c. During the dinner time period, the arrival rate is 1 per minute. What are your answers to (a) and (b) for this period?

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