#2: Suppose a random variable X has expected value E[X] = 3 and variance Var(X) = 4. Compute the following quantities. (a) E[3X+2] (b) E[X²] (c) E[(3X+2)²]
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![#2: Suppose a random variable X has expected value E[X] = 3 and variance Var(X) = 4. Compute the following
quantities.
(a) E[3X + 2]
(b) E[X²]
(c) E[(3X+2)²]
(d) Var[2X + 2]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F28b6003d-0204-46eb-90bf-e25d034fc92a%2Fdb2fb6ff-30ec-4b2e-aeca-eb5e14c4704e%2Fnrwcuvr_processed.png&w=3840&q=75)
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- 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.If X is a Poisson variable such that P(X =2) = 9P(X= 4) + 90P(X = 6), find the mean and variance of X.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]
- 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]?(b) Find the variance Var(X) of X.f(X)= 3/8( 4x - 2x2 ) 0<* x <* 2 (* less or equal to) a. Find the variance of X.
- If X is a Poisson variable such that P(X = 2) = 9P(X = 4) +90 P(X = 6), find the mean and variance of X.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]The 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) k
- The 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] "Suppose X and Y are independent. X has a mean of 1 and variance of 1, Y has a mean of 0, and variance of 2. Let S=X+Y, calculate E(S) and Var(S). Let Z=2Y^2+1/2 X+1 calculate E(Z). Hint: for any random variable X, we have Var(X)=E(X-E(X))^2=E(X^2 )-(E(X))^2, you may want to find EY^2 with this. Calculate cov(S,X). Hint: similarly, we have cov(Z,X)=E(ZX)-E(Z)E(X), Calculate cov(Z,X). Are Z and X independent? Are Z and Y independent? Why? What about mean independence?b. Find E[X(X – 1)] for Poisson random variable. Use the result to find the variance of X.
