1. The diameter of a particle of contamination (in micrometers) is modeled with the probability density function f(x)=2/(x^3) for x>1. Determine P(X<4 or X>8). O
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- Consider the functionf(x)=1/x, defined on the interval 1≤ x≤ e describe the shape of distribution after graphing the density function.If X and Y are independent RVs each normally distributed with mean zero and -1 Y variance o, find the density functions of R = X² + Y² and o = tan X %3DThe shelf life (in days) for bottles of a product is a continuous random variable having the density function 1 ifx>1 f(x)={ x2' 0, else. Let F(x) be the cumulative distribution function for X. Find F(7.85). Round your answer to a number with two decimal digits after the decimal point. For example if your answer is 1/40, which is equal to 0.025, then you should enter 0.03.
- The number of minutes that train is early or late can be modeled by a random variable whose density is given by: g(t) = {(1/972)(81 − t^2), −9 ≤ t ≤ 9, 0 elsewhere, where negative values indicate the train arriving early and positive values indicate the train arriving late. a. Find the probability that one of the train trips will arrive more than 5 minutes early. b. Find the probability that one of the train trips will arrive between 1 and 8 minutes late. c. Without integration, give the expected number of minutes early/late. Examination of the graph and recollection of properties of integrals will allow this.Hi. Please help. I need to calculate the mean and vairance of the probabiltiy density function I am getting one minute but the time better each serving is at least 2 minutes. Can you correct me as I have gone wrong somewhere. (see attached doc)The lifetime, X, of a particular integrated circuit has an exponential distribution with rate of λ=0.5 per year. Thus, the density of X is: f(x,x) = 1 e-^x for 0 ≤ x ≤ ∞o, λ = 0.5. λ is what R calls rate. Hint: This is a problem involving the exponential distribution. Knowing the parameter for the distribution allows you to easily answer parts a,b,c and use the built-in R functions for the exponential distribution (dexp(), pexp(), qexp()) for other parts. Or (not recommended) you should be able to use the R integrate command with f(x) defined as above or with dexp() for all parts. d) What is the probability that X is greater than its expected value? e) What is the probability that X is > 5? f) What is the probability that X is> 10? g) What is the probability that X > 10 given that X > 5? h) What is the median of X? Please solution USING R script
- Find the probability that solar radiation will exceed 400 calories on a typical October de 5.7 The distribution function of a random variable X is as follows: 0. 0 2 ' A. Graph the distribution function. b Find the probability that X is between 0.25 and 0.75. Find the probability density function of X. Kd Graph the probability density function of X.2. In probability, it is common to model the deviation of a day's temperature from the monthly average temperature using the Gaussian probability density function, 1.-/9 f(t) = V9T This means that the probability that the day's temperature will be between t = a and t = b different from the monthly average temperature is given by the area under the graph of y = f(t) between t a andt = b. A related function is %3D F(x) = -2/9 dt, 2 0. %3D V9n Jo This function gives the probability that the day's temperature is between t = -x and t = x different from the monthly average temperature. For example, F(1) 0.36 indicates that there's roughly a 36% chance that the day's temperature will be within 1 degree (between 1 degree less and 1 degree more) of the monthly average. 1 (a) Find a power series representation of F(x) (write down the power series using sigma notation). (b) Use your answer to (a) to find a series equal to the probability that the day's temperature will be within 2 degrees of the…6.106 If Y is a continuous random variable with distribution function F(y), find the probability density function of U = F(Y).
- A given continuous distribution has a probability density function: f(x)=7e^-(x/3a) where 0 ≤ x ≤1.5 I. Determine the value of aa for the given probability density function. ii. Determine the cumulative density function iii. With the given probability distribution, find P(0.1≤X≤0.3)2. Assume that if a certain scale in a laboratory has been used too long without recalibration the probability density function of the measurement error is f(x) = 1 – 0.5x for 022