Problem #2: Suppose a random variable X has expected value E[X]=-2 and variance Var(X) 4. Compute the following quantities. (a) E[X-2] (b) E[X²] (c) E[(x-2)²] (d) Var[X-3]
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- Repeat Example 5 when microphone A receives the sound 4 seconds before microphone B.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 X and Y be identically distributed with mean 4 and variance 25.(a) If X and Y are uncorrelated, what is the variance of X + Y?(b) If the covariance between X and Y is 8, what is the variance of X + Y? (c) If X and Y are independent, what is E[(X - Y)^2]?
- Suppose a random variable X has expected value E[X] = -4 and variance Var(X) = 16. Compute the following quantities. (a) E[X-1] (b) E[X²] (c) E[(x - 1)²] (d) Var[3X + 2]Suppose a random variable X has expected value E[X] = -1 and variance Var(X) = 16. Compute the following quantities. (a) E[2X-3] (b) E[X²] (c) E[(2x - 3)²] (d) Var[2X - 2]If X is a random variable with pdf f(x) = 2x − 2 where x = (1, 2), find the variance of Y = 2X - 3.
- Suppose that the probability that a patient admitted in a hospital is diagnosed with a certain type of cancer is 0.03. Suppose that on a given day 10 patients are admitted and X denotes the number of patients diagnosed with this type of cancer. The mean and the variance of X are: None of these E(X)=0.5 and V(X)=0.475 E(X)=0.4 and V(X)=0.384 E(X)=0.3 and V(X)=0.291The p.d.f. of a random variable X' is as shown in the figure. The pdf is zero for X 5. Calculate (i) the maximum value of p.d.f. (ii) expectation of X, E(X) (iii) variance of X. fx (x) kThe random variable X takes value O with probability, and value 1 with the remaining probability. The random variable Y takes value 100 with probability and value 101 with the remaining probability. What is the relation between the variance of X and the variance of Y? Var[X] Var[Y] "