25. (a) Define the Integral of a simple random variable and provide an illustrative example.

An ordinary (fair) coin is tossed 3 times. Outcomes are thus triple of “heads” (h) and tails (t) which we write hth, ttt, etc. For each outcome, let R be the random variable counting the number of tails in each outcome. For example, if the outcome is hht, then R (hht)=1. Suppose that the random variable X is defined in terms of R as follows X=6R-2R^2-1. The values of X are given in the table below. A) Calculate the values of the probability distribution function of X, i.e. the function Px. First, fill in the first row with the values X. Then fill in the appropriate probability in the second row.

Let X be a strictly positive random variable. Determine which of the following quantitiesare larger:(a)  E(X^4) or (E(X))^4(b)  E(X^1/9) or (E(X))^1/9(c)  E(X^3 + 4X^2 + 9) or (E(X))^3 + 4(E(X))^2 + 9(d)  E(e^X ) or e^E(X)(e)  E(arctan(X)) or arctan(E(X))

Find the variance of the random variable X with a PDF given by 3x2 S* if - 1< x< 1, f(x) = 2 otherwise. Submit your answer in fractional form.

Suppose that Y is a uniform continuous random variable on the interval (1,11). Calculate the expected value of the random variable (Y− 1)². Use one decimal place accuracy.

For each characteristic, decide whether the random variable is continuous or discrete. (a) The number of left-handed individuals in a sample of 250 people. O continuous O discrete (b) Time taken to complete an exam for students in a class. O continuous O discrete (c) Vehicle speeds at a highway location. O continuous O discrete (d) The number of accidents reported last year at a highway location. O continuous O discrete

The weekly demand for propane gas (in 100's of gallons) from a particular facility is a random variable X with pdf: f(x) = 2(1) for 1 ≤ x ≤ 2 Please find out the mean of X. 3/2-In(2) 3-2 In(2) 1/2-In(4) 3-In(2)

are iid random variables having mean u and variance o?. Form the random Suppose f1, 2, . sum SN .. E1 +..+ EN. (a) Derive the mean and variance of SN when N has a Poisson distribution with parameter d. (b) Determine the mean and variance of SN where N has a geometric distributions with mean (c) Compare the behaviors in (a) and (b) as A→ o.

Let Z be nomrally distributed with mean 0 and variance 1. Let Z1 ,.., Zn be an iid random sample, all distributed as Z.  Find the moment-generating function of Z2. please show all your work especially when intergrating. do not skip any minor details.

True or False: For an exponentially distributed random variable, x, P(x) = 1/(b - a).