The following table shows the distribution of the digits 0, 1, 2,….,9 in a random table of 250 digits. Does the observed distribution differ significantly from the uniform distribution at 0.05 level of significance?
| Digit | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
| Observed frequency | 17 | 31 | 29 | 18 | 14 | 20 | 35 | 30 | 20 | 36 |
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If a random variable U has the t-distribution with n degrees of freedom, show that Z = U2 has the F-distribution with 1 and n degrees of freedom.
hich of the following statements are true or false? Give reasons for your answer.
i) If P(A) = 0.5, P(A È B) = 0.7 and A and B are independent events, then P(B) = .
ii) There exists a random variable X for which
.
iii) If X has the distribution N(10, 9) and if Y = 5X + 3, then Y has distribution N(53,81).
iv) The abscissa of the point of intersection of two gives of less-than type and more- than type gives the mean of the data.
v) In a problem of testing of a simple hypothesis against a simple alternative, if the probability of type 1 error is known to be 0.06, then the power of the test will be 0.94.
For the following frequency table obtain the relative frequencies and the cumulative frequencies of the ‘more than’ type.
| Life (in hours)
499.5 – 649.5 | Frequency
5 |
| 649.5 – 799.5 | 20 |
| 799.5 – 949.5 | 40 |
| 949.5 – 1099.5 | 81 |
| 1099.5 – 1249.5 | 57 |
| 1249.5 – 1399.5 | 34 |
| 1399.5 – 1549.5 | 13 |
| Total | 250 |
Let X be a r.v. with p.m.f. given by the following table:
| - 2 | - 1 | 0 | 1 | 2 |
| 3 20 | 4 20 | 6 20 | 4 20 | 3 20 |
Let the p.d.f. of a continuous random variable X be given by
Find the maximum likelihood estimator of based on a sample X1 ,X2 ,....,Xn .
In a certain community, 10% of all people above 50 years of age have diabetes. A health service in this community correctly diagnoses 95% of all persons with diabetes as having the disease, and incorrectly diagnoses 5% of all persons without diabetes as having the disease. Find the probability that a person randomly selected from among all people of age above 50 and diagnosed by the health service as having
diabetes actually has the disease.
Suppose X has the density function
Find its moment generating function whenever it exists.