Let Xn be independent exponentially distributed random variables with means un = µ + 1/n with μ > 0. Then show the weak low of large numbers applies, such that (X1 + ··· + Xn)/n ->pμ
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- Find the average rates of change of f(x)=x2+2x (a) from x1=3 to x2=2 and (b) from x1=2 to x2=0.Respiratory Rate Researchers have found that the 95 th percentile the value at which 95% of the data are at or below for respiratory rates in breath per minute during the first 3 years of infancy are given by y=101.82411-0.0125995x+0.00013401x2 for awake infants and y=101.72858-0.0139928x+0.00017646x2 for sleeping infants, where x is the age in months. Source: Pediatrics. a. What is the domain for each function? b. For each respiratory rate, is the rate decreasing or increasing over the first 3 years of life? Hint: Is the graph of the quadratic in the exponent opening upward or downward? Where is the vertex? c. Verify your answer to part b using a graphing calculator. d. For a 1- year-old infant in the 95 th percentile, how much higher is the walking respiratory rate then the sleeping respiratory rate? e. f.Let X1 and X2 be IID exponential with parameter > 0. Determine the distribution ofY = X1=(X1 + X2).
- Assume that the entire sample has 2 positive observations and 6 negatives observations. Variable X1: at the left branch has 2 positive observations and 4 negatives observations;at the right branch has 6 positive observations and 6 negatives observations. What is the information gain or reduction in uncertainty of X1 using the Gini index? (round to two decimal spaces)Assume δ = 0.05, ̄Ax = 0.06, 2 ̄Ax = 0.0040. A group of 100 lives age x purchasefully continuous whole life policies of face value $1,000 for continuous premiums of $3.50per year. Let L be the aggregate loss function (L =100∑i=1Li, where Li represents the netfuture loss of individual i). Calculate the mean and variance of the aggregate loss L.Show that variance o? = (x²) – ((x))²
- Assume that the entire sample has 8 positive observations and 4 negatives observations. Variable X1: at the left branch has 9 positive observations and 3 negatives observations;at the right branch has 8 positive observations and 8 negative observations. What is the information gain or reduction in uncertainty of X1 using the Gini index? (round to two decimal spaces)Suppose that the random change in value of a financial asset is X over the first day and Y over the second. Suppose also that Var(X) =18 and Var(Y) = 26 In this case, the total change in the value over these two days is given by X +Y. Do you have enough information to compute Var(X +Y)? If so, compute this value. If not, explain what additional information you need to do so.The time (in hours) required to repair a machine is exponentially distributed 1 with parameter 2 = = what is the probability that the repair time exceeds 3 3