Suppose that X is a continuous random variable with the following probability densityfunction:f (x) = 1θ , 0 ≤ x ≤ θJimmy considers himself a budding statistician and he wants to investigate the true valueof θ.(a) [2 marks] Jimmy thinks that the true value is actually θ = 2. Assuming Jimmyis right, find each of the following probabilities:P(0 < X < 13), P(13 < X < 1), P(1 < X < 74), P(74 < X < 2)(b) [4 marks] In order to test whether θ = 2, Jimmy collects a sample of 500 of theseX variables and records their values. He then tabulates how many variables fellinto each range of part (a), and his results are summarised below. Based on thisdata, test whether θ = 2. Clearly state your hypotheses and use a significance levelof α = 5%.Range of value (0 < X < 13) (13 < X < 1) (1 < X < 74) (74 < X < 2)Number of variables 101 165 191 43(c) [4 marks] Jimmy now wants to actually estimate θ. From the sample he collectedin part (b), he can calculate the sample mean, ¯X =∑500i=1 Xi500 . Is ¯X an unbiasedestimator of θ? Why or why not? If not, derive an unbiased estimator of θ.(d) [3 marks] Jimmy decides it might be a better idea to use an interval estimatorrather than a point estimator. Based on the sample mean of ¯X = 0.9418 andthe population variance of σ2 = 0.3008, calculate a 95% confidence interval forµ = E(X). Interpret this confidence interval.(e) [2 marks] Without actually performing the test, if you were to test H0 : µ = 1against the two-tailed alternative at a significance level of α = 5%, would you rejectH0? Why or why not?(f) [2 marks] Jimmy actually wanted an interval estimator for θ, not µ. Suggest a wayto convert the confidence interval for µ you constructed in part (d) to a confidenceinterval for θ.

Suppose X is a continuous random variable with the following probability density func-tion:f (x) = c(2 − x), 0 ≤ x ≤ 2(a) [2 marks] Plot the probability density function, clearly marking on your plot thevalues of f (x) when x = 0 and x = 2 and labelling both axes.(b) [2 marks] Explain and show why the value of c must equal 12 .(c) [2 marks] Find P (1 < X < 1.5).(d) [3 marks] Find P ({1 < X < 1.5}|{X > 0.5}).(e) [2 marks] Find P ({X > 0.5}|{1 < X < 1.5}).(f) [4 marks] Suppose that we have another random variable U that is uniformlydistributed with parameters 2 and 4. That is, the probability density function of Uis given byf (u) = 12, 2 ≤ u ≤ 4.Further, U and X are independent. In addition, someone has told us that therandom variable Y defined byY = X + Uhas expected value equal to 113 and variance equal to 59 . Based on this information,calculate the expected value and variance of X

What is the Probability Density Function (PDF) of the continuous reward variable X in this RLenvironment?

Cumulative Probability of Continuous VariablesSuppose you work at a sports analysis company and you want to analyse the effect a bowler’s height has on his/her performance. So, you create a list of all 5 wicket hauls in the last decade. Based on this data, they created a cumulative probability distribution for X, where X = height of the bowler who took the 5 wicket haul.Now, based on the data, you conclude that the cumulative probability, F(175.3 cm) = 0.3. In this case, which of the following statements is correct?P(X<175.3 cm) = 0.3P(X<175.3 cm) = 0.3(Remember that height is a continuous variable.)

LetX = (X1, . . . , Xn)Tdenote a random sample of size n on a discrete random variable X which hasprobability function f (x; θ), where θ is an unknown parameter.Suppose we wish to estimate f (k; θ), the probability that X is equal to agiven arbitrary (non-negative) integer k. We shall subsequently write f (k; θ)as the function g(θ), suppressing its dependence on the value k.Let U (X1) be a statistic defined as:U (X1) ={1, X1 = k,0, otherwise.(i) On letting T (X1, ..., Xn) denote a sufficient statistic for θ, show thatW (T ) = E{U (X1) | T (X1, ..., Xn)}is a statistic; that is, explain why it does not depend on θ.(ii) Show that W (T ) is an unbiased estimator of g(θ).(iii) Without assuming the result of any theorem, show that the variance ofW (T ) will be less than the variance of U unless U is constant given Twhere in this case W will have the same variance as U .(iv) Explain why W (T ) is the UMVU estimator of g(θ) in the case whereT is a complete, minimal sufficient statistic for θ.

You work at a sports analysis company and want to analyse the effect a bowler’s height would have on their performance. So, you created a list of all five-wicket hauls in the last decade. Based on this data, a cumulative probability distribution for X was created, where X = height of the bowler who took the five-wicket haul.Based on the data, you have concluded that the cumulative probability, F(175.3 cm) = 0.3. In this case, which of the following statements is correct?P(X < 175.3 cm) = 0.3P(X < 175.3 cm) = 0.3Note: Remember that height is a continuous variable.Only statement 1 is correct.Only statement 2 is correct.Both statements 1 and 2 are correct.