Example Determine the probability density of the length of a radius- vector if the coordinates of its end A obey the normal circular distribution law + f(x, y) = 2m ² exp{- *²2222²}. 2TT
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- Let T have an exponential distribution with parameter λ. Let Y = √T .a) Find the density of Yb) Find the expectation of Y, correct to two decimal places, for λ = 3.c) A random number generator produces uniform [0, 1] random numbers. How could you usethese to generate random numbers which have the distribution of Y ?The probability density function of the random variable X is as in the picture with λ> 0. Find the moments estimator (λ^) of the parameter λ.The random variable X, the particle size (in micrometers) distribution is characterized by the probability density function:f(x) = 3x^(-4) , x > 1 and 0 elsewhereFind the probability that X is less than 3.8 micrometers?
- Suppose that the random variable B has the standard normal density. What is the conditional probability density function of the sum of the two roots of the quadratic equation x2 + 2Bx + 1 = 0 given that the two roots are real? KINDLY REQUEST YOU TO PROVIDE ME WITH COMPLETE SOLUTIONThe random variable X, the particle size (in micrometers) distribution is characterized by the probability density function:f(x) = 3x^(-4) , x > 1 and 0 elsewhereFind the probability that X exceeds 1.8 micrometers?Let x be a continuous random variable with the density function: f(x) = 3e-3x when x>0 and 0 else Find the variance of the random variable x.
- If X has the uniform density with the parameters α = 0and β = 1, use the distribution function technique to findthe probability density of the random variable Y = √X.The probability density function of the random variable X is as in the picture with λ> 0. Find the maximum likelihood estimator (λ^) of the parameter λ.The joint probability function of random variables X and Y is given to be: (in the picture) a. Find the marginal density functions of X and Y b. Prove the independency of the variables
- Suppose that the lifetime X (in hours) of a certain type of flashlight battery is a random variable on the interval 30 ≤ x ≤ 50 with density function f(x) = 1/20, 30 ≤ x ≤ 50. Find the probability that a battery selected at random will last at least 35 hours.he length of time, in minutes, for an airplane to obtain clearance for takeoff at a certain airport is a random variable Y = 3X-2, where X has the density function. Find the mean and variance of the random variable Y.On a production line, parts are produced with a certain average size, but the exact size of each part varies due to the imprecision of the production process. Suppose that the difference between the size of the pieces produced (in millimeters) and the average size, which we will call production error, can be modeled as a continuous random variable X with a probability density function given by f(x) = 2, 5e^(-5|x|), for x E R (is in the image). Parts where the production error is less than -0.46 mm or greater than 0.46 mm should be discarded. Calculate (approximating to 4 decimal places): a) What is the proportion of parts that the company discards in its production process? b) What is the proportion of parts produced where the production error is positive? c) Knowing that for a given part the production error is positive, what is the probability of this part being discarded?