JNTU
JNTU B.Tech II COMPUTER ORIENTED STATISTICAL METHODS Previous Question Paper (March - 2021 - R18)
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Main
Two dice are thrown the random variable is assigned to the sum. Write the distribution. Find the mean and variance.
Answer
When two fair dice are thrown, there are 6 × 6 = 36 equally likely outcomes. Let X be the random variable representing the sum of the two dice. The possible values of X range from 2 to 12. The probability distribution P(X=x) is calculated by counting the number of ways to obtain each sum:
| x | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
| P(X=x) | 1/36 | 2/36 | 3/36 | 4/36 | 5/36 | 6/36 | 5/36 | 4/36 | 3/36 | 2/36 | 1/36 |
The mean (expected value) E[X] is calculated as:
To find the variance, we first calculate E[X²]:
The variance Var(X) is given by:
If the probability distribution function of a continuous random variable is defined as
Answer
i) To find k, we use the property that the total area under the probability density function must be 1:
Since the function is symmetric about x=0, we can write:
ii) The mean E[X] is given by:
Since x e-|x| is an odd function, the integral over the symmetric interval (-∞, ∞) is 0. Thus, the mean is 0.
iii) The variance V[X] is given by E[X2] - (E[X])2. Since E[X] = 0, V[X] = E[X2]:
Using the gamma function integral formula ∫0∞ xn e-ax dx = n! / an+1 with n=2 and a=1:
A sample of 4 items is selected from 12 out of which 5 are defective. Find the expected number of defective items.
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Eight coins are tossed. Find the probability of getting heads: i) P(X = 3) ii) P(X ≤ 4).
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The probabilities of a Poisson variate taking the values 1 and 2 are equal. Calculate: i) P(X = 0) ii) P(X = 3).
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Mean heights of students is 159cms with a standard deviation of 20. Find how many students heights lie between 150cms and 170cms in a class of 100 students.
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The expected number of typographical errors on a page of a certain magazine is 0.2. What is the probability that the next page you read contains i) 0 and ii) 2 or more typographical errors?
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From the following data find whether there is any significant liking in the habit of taking soft drinks among the categories of employees.
| Soft drinks | Clerks | Teachers | Officers |
|---|---|---|---|
| Pepsi | 10 | 25 | 65 |
| Thumsup | 15 | 30 | 65 |
| Maaza | 50 | 60 | 30 |
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Two horses A and B were tested according to the time (in seconds) to run a particular track with the following results. Test whether two horses have the same running capacity.
Horse A: 28, 30, 32, 33, 33, 29, 34
Horse B: 29, 30, 30, 24, 27, 29
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A random sample of 100 electric bulbs, produced by a manufacturer A showed a mean life of 1190 hrs with a standard deviation of 90. Another sample of 75 electric bulbs produced by a manufacturer B showed a mean life of 1230 with a standard deviation of 120 hrs. Find whether there is significant difference between the mean.
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50 people were attacked by a disease and 30 were survived. If the survival rate is 70%, test the chain at 5% level.
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Consider a three-state Markov chain with the transition matrix P = [[0, 1, 0], [0.2, 0.3, 0.5], [0.16, 0.15, 0.16]]. If the initial probabilities P0 = (0.2, 0.3, 0.5).
Find the probabilities after two transitions.
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Find the limiting probabilities.
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