1 e length of a petal on a certain flower varies from 2.56 cm to 6.76 cm and has a probability density function defined by f(x) = Find the probabilities that the length of a randomly selected petal will be as 2/x Ows. Greater than or equal to 3.4 cm Less than or equal to 3 cm Between 3 cm and 3.4 cm

The graph to the right is the uniform probability density function for a friend who is x minutes late (a) Find the probability that the friend is between 5 and 25 minutes late (b) it is 10 AM. There is a 30% probability the friend will arrive within how many minutes? (a) The probability that the friend is between 5 and 25 minutes late is (Type an integer or a decimal Round to three decimal places as needed.) (b) There is a 30% probability the friend will arrive within minutes (Type a whole number) Ang 100 10 20 30x Tem

Suppose that the interval between eruptions of a particular geyser can be modelled by an exponential distribution with an unknown parameter 0 > 0. The probability density function of this distribution is given by f(x; 0) = 0e 0¹, x > 0. The four most recent intervals between eruptions (in minutes) are x₁ = 32, x₂ = 10, x3 = 28, x4 = 60; their values are to be treated as a random sample from the exponential distribution. (a) Show that the likelihood of based on these data is given by L(0) 04-1306 = (b) Show that L'(0) is of the form L'(0) = 0³ e 1300 (4- 1300). (c) Show that the maximum likelihood estimate of 0 based on the data is ~ 0.0308 making your argument clear. (d) Explain in detail how the maximum likelihood estimate of that you have just obtained in part (c) relates to the maximum likelihood estimator of for an exponential distribution.

2. Consider a gas station's daily sales. Let Y be the volume of gas sold per day in 1000s of gallons. Assume the distribution of Y follows this pdf: 1sxs 2 S (v) . else What value of k is required for the above function to be a valid pdf? b. Verify that the function given is a valid probability density а. function c. Determine the cumulative distribution function (cdf) of the random variable d. Compute the probability that there is exactly 1500 gallons of sales on a given day (Hint: Find?(r - 1.5)) = 1 CO0