There are 2 categories: category 1 follows a uniform distribution: Sk (0 < x < 2), f(x) = 10 (otherwise) category 2 follows an exponential distribution: 1 f(x) = e-*(x > 0) %3D - m If the prior for both categories are the same. Determine the decision boundaries (assume x>0) Hint: determine k and m first.

Suppose that a pair of fair dice are to be tossed, and let the random variableX denote the sum of the points. Construct a table showing the probabilitydistribution for X. Find the distribution function F(x). Plot the graph of thedistribution function. Find (i ) P(X < 12) and (ii) P(X = 3)

A hospital is testing patients for Covid daily. The probability that a patient has covid is 34%. What is the probability that the 1st patient to test positive for covid will be found in the fifth test? Calculate V(X) and show distributions. show steps and work

distribution table

Let X-T(4) and Y~ x2 (4). The following plot depicts the cumulative distribution functions of X and Y. We don't know which is which. The horizontal lines have values 25% and 75%. We denote by q1 the x-coordinate correponding to the red point and by q2 the x-coordinate corresponding to the blue point. Which of the following statements are true? 1.00 - 0.75 0.50 - 0.25+ 0.00 - The answer is : The cdf of X corresponds to the red curve while the one of Y corresponds to the blue curve. Can you explain why? Density

The following table is a partial probability distribution for the MRA Company's projected profits (x = profit in $1,000s) for the first year of operation (the negative value denotes a loss). x    f(x) −100    0.100    0.2050    0.30100    0.25150    0.10200    (a) What is the proper value for f(200)? f(200) = What is your interpretation of this value? This is the  ---Select--- percent probability that the MRA will have a $200,000 profit. (b) What is the probability that MRA will be profitable? (c) What is the probability that MRA will make at least $100,000?   Please only correct and with concept and dont do handwritten or skip please!!

A researcher claims that the number of homicide crimes by season is uniformly distributed. To test this claim, you randomly select 1,201 homicides from a recent year and record the season when each happened. Season Frequency, f The table shows the results. At a = 0.01, test the researcher's claim. Spring Summer 305 316 Fall 299 Winter 281 State H, and H, and identify the claim. Ho: The distribution of the number of homicide crimes by season H,: The distribution of the number of homicide crimes by season Which hypothesis is the claim? O Ho O Ha Calculate the test statistic. x2 = (Round to three decimal places as needed.) Determine the P-value. P-value =(Round to three decimal places as needed.) Decide whether to reject or fail to reject the null hypothesis and interpret the decision in the context of the original claim. Hp. There enough evidence at the 1% level of significance to reject the claim that the distribution of the number of homicide crimes by season

PI. A shipment of 8 television sets contains 3 defective sets. A hotel makes a random purchase of 4 these sets. If X is the number of defective sets purchased by the hotel, find (a0 the probability distribution and (b) the cumulative distribution of X. Using F(x), find (c) P(X = 2) and (d) P(0 < X< 3)

5. Using the appropriate distribution function for each, evaluate the following. (The answer is an algebraic expression): i. (v³) ii. (v)

Q3. Exponential Distribution has a memoryless property. Intuitively, it means that the probability of customerservice answering you call (assuming waiting time is exponential) in the next 10 mins is the same, no matter ifyou have waited an hour on the line or just picked up the phone. Formally, if X ∼ exponential(λ), f(x) = λ exp(-λx), and t and s are two positive numbers, use the definition of conditional probability to show thatP(X > t + s | X > t) = P(X > s).Hint: Find the cdf of X first, and note that P(X > t + s Ç X > t) = P(X > t + s)