If x has an exponential distribution with parameter θ . Find the density function of
let's find the density function of the random variable \( Y = \ln(X) \), where \( X \) has an exponential distribution with parameter \( \theta \).
The exponential density function for \( X \) is given by:
\[ f_X(x) = \begin{cases} \theta e^{-\theta x} & \text{for } x \geq _____ ____ __________ ___ ______ _____.
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Let X be a single observation from the p.d.f.
If X ≥1 is the critical region for testing against the alternative hypothesis
obtain the values of type I and type II errors.
Draw the cumulative frequency curves for the following distribution:
| Marks | No. of Students |
| 0 – 10 | 4 |
| 10 – 20 | 8 |
| 20 – 30 | 11 |
| 30 – 40 | 15 |
| 40 – 50 | 12 |
| 50 – 60 | 6 |
| 60 – 70 | 3 |
The police plans to enforce speed limits by using radar traps of 4 different locations within the city. The radar traps at each of the locations and
are operated 40 % 30% 20% and 30%, of time. If a person who is speeding on h, is way to work has probabilities of 0.2, 0.1, 0.5 and 0.2 respectively of passing through these locations. What is the probability that the person will receive a speeding challan.
An unbiased die is rolled twice. Let denote the event: odd face roll on the first die,
denote the event that total score is Odd. Check the independence of
and .
Find the maximum likelihood estimator for the parameter λ of the Poisson distribution on the basis of a sample of size n . Also, find its variance.
Which of the following statements are True or False? Give reasons for your answer.
i) If then the correlation coefficient between x and y does not exist and if it exists it is equal to zero.
ii) For a normal distribution mean, median and standard deviation are all equal.
iii) If then
assume only non-positive values and hence
iv) If two unbiased dice are rolled, then the probability of their same score being 6 is
v) If the random variable x follows a normal distribution with known mean µ and unknown variance then
is a statistic but ) (x −µ is not.
A group of 250 items with mean 15.6 and standard deviation . has been divided into two groups. The first has 100 items with mean 15 and standard deviation 3. Find the standard deviation of the second group.