Question

Find the mean and variance of the random variable X with probability function or density $f(x)$. 1. $f(x) = kx$ ($0 \le x \le 2$, k suitable) 2. X = Number a fair die turns up 3. Uniform distribution on $[0, 2\pi]$ 4. $Y = \sqrt{3}(X - \mu)/\pi$ with X as in Prob. 3 5. $f(x) = 4e^{-4x}$ ($x \ge 0$) 6. $f(x) = k(1 - x^2)$ if $-1 \le x \le 1$ and 0 otherwise 7. $f(x) = Ce^{-x/2}$ ($x = 0$) 8. X = Number of times a fair coin is flipped until the first Head appears. (Calculate $\mu$ only.) 9. If the diameter X [cm] of certain bolts has the density $f(x) = k(x - 0.9)(1.1 - x)$ for $0.9 < x < 1.1$ and 0 for other x, what are k, $\mu$, and $\sigma^2$? Sketch $f(x)$.

          Find the mean and variance of the random variable X with probability function or density $f(x)$. 
1. $f(x) = kx$ ($0 \le x \le 2$, k suitable)
2. X = Number a fair die turns up
3. Uniform distribution on $[0, 2\pi]$
4. $Y = \sqrt{3}(X - \mu)/\pi$ with X as in Prob. 3
5. $f(x) = 4e^{-4x}$ ($x \ge 0$)
6. $f(x) = k(1 - x^2)$ if $-1 \le x \le 1$ and 0 otherwise
7. $f(x) = Ce^{-x/2}$ ($x = 0$)
8. X = Number of times a fair coin is flipped until the first Head appears. (Calculate $\mu$ only.)
9. If the diameter X [cm] of certain bolts has the density $f(x) = k(x - 0.9)(1.1 - x)$ for $0.9 < x < 1.1$ and 0 for other x, what are k, $\mu$, and $\sigma^2$? Sketch $f(x)$.

Find the mean and variance of the random variable X with probability function or density f(x).
1. f(x) = kx (0 ≤ x ≤ 2, k suitable)
2. X = Number a fair die turns up
3. Uniform distribution on [0, 2π]
4. Y = √(3)(X - μ)/π with X as in Prob. 3
5. f(x) = 4e^-4x (x ≥ 0)
6. f(x) = k(1 - x^2) if -1 ≤ x ≤ 1 and 0 otherwise
7. f(x) = Ce^-x/2 (x = 0)
8. X = Number of times a fair coin is flipped until the first Head appears. (Calculate μ only.)
9. If the diameter X [cm] of certain bolts has the density f(x) = k(x - 0.9)(1.1 - x) for 0.9 < x < 1.1 and 0 for other x, what are k, μ, and σ^2? Sketch f(x).

Added by Sheila P.

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