1. The time between Rajini movies are exponentially distributed random variables with unknown parameter L (units 1/months). Assume that the time between his movies are independent random variables. The time between his last 6 movies in months are 10, 14, 18, 8, and 20 months. What is the Maximum Likelihood Estimate for the unknown parameter L?
2. We are given i.i.d. observations X_1,X_2,...,X_n that are uniformly distributed over the interval [T,T+1].
Find a maximum likelihood estimator for T.
Is this estimator consistent?
Is this estimator unbiased?
Is this estimator asymptotically unbiased?
3. Let X be a normal random variable with mean 'm' and unit variance. We want to test the hypothesis 'm=5' at the 5% level of Significance, using n independent samples of X.
What is the range of values of the sample mean for which the hypothesis is accepted?
Let n=10. Calculate the probability of accepting the hypothesis 'n=5' when the true value of the mean is '4'.
We have five observations drawn independently from a normal distribution with unknown mean 'm' and variance 'v'
Estimate 'm' and 'v' if the observation values are 8.47, 10.91, 10.87, 9.46, 10.40
Use the t-distribution tables to test the hypothesis m=9 at the 95% significance level, using the estimates of previous part
Accept the hypothesis
Reject the hypothesis
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