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- A probability distribution function is modeled by the following X-Y cartesian graph: 15k 长 6 7 12 Determine the following: 1. Probability density function for the intervals defined on the graph. 2. Expected value of X. 3. Variance of X. I 4. P(X<5) 5. P(2The probability density function of the random variable X is given by 3. f(2) = , 0 Sz<2 otherwise 8 Calculate the variance Var(X). Enter your answer numerically and accurate to 2 decimal places. For example, if the value you calculated is 0.998, you should enter 1.00; if the value you calculated is 0.941, you should enter 0.94.3) Find the mean and variance of the random variable X with probability function or density f(x). (a)f(x)={k(1-x), –15xs1 O otherwise (b) ƒ(x) = k/3× , x = 1,2,3,... Note: in this problem, you need to first find the values of k.Let the random variable X have the density function is given by F(e) =*. Os s5, lo, elsewhere. 1) The value of the constant C=; 2) The expected value of X is E(X)=; 3) The expected value of X2 is E (x2) =; 4) The variance of X is given by Var (X) =o? =.Example 3: The probability density ƒ(x) of a continuous random variable is given by f(x) 3 ce , -m9. Zero-mean Gaussian random variables x and y have variance oz= 3 and o= 4 respectively and a y -1 correlation coefficient p = ing Write an expression for the joint density function.Suppose that f (x) = 0.125x for 0 < x < 4 Determine the mean and the variance of X.3. The level of impurity (in percent) in the product of a certain chemical process is a random variable with probability density function 3 64 -x²(4 - x) 0 0 < x < 4 otherwise a. What is the probability that the impurity level is greater than 3%? b. What is the probability that the impurity level is between 2% and 3%? c. Find the mean impurity level. d. Find the variance of the impurity levels.Y is a continuous random variable with the probability density function pictured. what is the mean of y?The total number of hours, in units of 100 hours, that a family runs a vacuum cleaner over a period of one year is a random variable X having the density function shown to the right. Find the variance of X (x-8). f(x)2-(X-8). 9sx< 10, 8* Let E(X]Y = y) = 3y and var(X|Y = y) = 2, and let Y have the p. d. f. se-y if y > 0 fG) = {o otherwise Then the variance of X isRecommended textbooks for youA First Course in Probability (10th Edition)ProbabilityISBN:9780134753119Author:Sheldon RossPublisher:PEARSONA First Course in Probability (10th Edition)ProbabilityISBN:9780134753119Author:Sheldon RossPublisher:PEARSON