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- Find the probability distribution of the random variable y, which represents the number of times writing appears when a balanced coin is tossed three times. Calculate the mean and variance of the random variable y, whose probability mass function is: f (y) = y^2/30 y = 1,2,3Let X be a random variable with PDF:According to the Maxwell–Boltzmann law of theoret-ical physics, the probability density of V, the velocity of a gas molecule, isf(v) =⎧⎨⎩kv2e−βv2for v > 00 elsewhere where β depends on its mass and the absolute tem-perature and k is an appropriate constant. Show that the kinetic energy E = 1 2mV2, where m the massof the molecule is a random variable having a gammadistribution.
- Suppose that X is a random variable with density and suppose that ?=e(-4X) Determine ?(?)The life X (in hours) of a battery in constant use is a random variable with exponential density. What is the probability that the battery will last more than 12 hours (h) if the average life is 8 h?Integrate the appropriate chi-square density to findthe probability that the variance of a random sample ofsize 5 from a normal population with σ2 = 25 will fallbetween 20 and 30.
- If a dealer’s profit, in units of $5000, on a new automobile can be looked upon as a random variable X having the density function Find the variance of X.An n sided die is rolled. Each face has a unique label from 1 to n. Let, X be therandom variable that denotes the number that is on top of the die after it haslanded. Express the variance of the variable X in terms of n.A boy is to sell lemonade to make some money to afford some holiday shopping. The capacity of the lemonade bucket is 1. At the start of each day, the amount of lemonade in the bucket is a random variable X, from which a random variable Y is sold during the day. The two random variables X and Y are jointly uniform. a) Write down the joint pdf of X and Y and clearly indicate the region of interest. b) What is the probability that the amount of lemonade sold is less than 1/3? c) Find and sketch the CDF and the pdf of "Z" which is the amount of lemonade remaining at the end of the day. Clearly indicate the range of Z
- let X and Y be independent standard normal random variables. Determine the pdf of W = X^2 + Y^2. Find the mean and the variance of U= (1/2)WThe Poisson mass function is often used in biology to model the number of bacteria in a solution. Suppose a dilute suspension of bacteria is divided into several different test tubes. The number of bacteria x in a test tube has a Poisson mass function with a parameter 2 that represents the mean number of bacterial cells contained in the different test tubes. Express the probability that a particular test tube contains no bacteria, in terms of 2.Show that the mean of a random sample of size n from an exponential population is a minimum variance unbiased estimator of the parameter 0.