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3 years ago

10

Evaluate and answer in standard form:4^2/3^2

1 answer:

Semenov [28]3 years ago

6 0

the answer is : 1.7777

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The PMF of $X$ is almost geometric in nature. Let $p=\frac{11}{29}$ . Then

$P(X = x) = \begin{cases} p & \text{if }x = 0 \\ (1-p)p & \text{if }x = 1 \\ (1-p)^2 p & \text{if }x = 2 \\ (1-p)^3p & \text{if }x = 3 \\ (1-p)^4 & \text{if }x = 4 \\ 0 & \text{otherwise}\end{cases}$

Compute the first moment/expected value.

$E(X) = \displaystyle \sum_x x\, P(X=x) \\\\ ~~~~~~~~ = 0\cdot p+1\cdot(1-p)p + 2\cdot(1-p)^2p + 3\cdot(1-p)^3p + 4\cdot(1-p)^4 \approx 1.39349$

Compute the second moment.

$E(X^2) = \displaystyle \sum_x x^2\, P(X=x) \\\\ ~~~~~~~~ = 0\cdot p+1\cdot(1-p)p + 4\cdot(1-p)^2p + 9\cdot(1-p)^3p + 16\cdot(1-p)^4 \approx 4.01103$

Compute the variance.

$V(X) = E(X^2) - E(X)^2 \approx 2.06921$

The standard deviation is the square root of the variance.

$\sqrt{V(X)} \approx \boxed{1.43848}$

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2 years ago