Question

Suppose that certain bolts have length L=400+X mm, where X is a random variable with density f(x)=3/4(1-x²) if -1<=x<=1 and 0 otherwise. Determine c so that with a probability of 95% bolt will have the length between 400-c and 400+c.

Answer 1

Answer on Question #57607 – Math – Statistics and Probability

Question

Suppose that certain bolts have length $L = 400 + X$ mm, where $X$ is a random variable with density

$$f(x) = \frac{3}{4}(1 - x^2) \text{ if } -1 \leq x \leq 1 \text{ and } 0 \text{ otherwise}.$$

Determine $c$ so that with a probability of $95\%$ bolt will have the length between 400-c and 400+c.

Solution

$$f(x) = \begin{cases} \frac{3}{4}(1 - x^2), & -1 \leq x \leq 1 \\ 0, & \text{otherwise} \end{cases}$$ $$\int_{-c}^{c} f(x) \, dx = 0.95.$$ $$\int_{-c}^{c} f(x) \, dx = \int_{-c}^{c} \frac{3}{4}(1 - x^2) \, dx = \frac{3}{4}\left(x - \frac{x^3}{3}\right)_{-c}^{c} = \frac{3}{4}\left(2c + \frac{(c)^3 - (-c)^3}{3}\right) = \frac{1}{2}(c + c^3) = 0.95$$

The solution of this cubical equation is

$$c = 0.974517.$$

Answer: 0.974517.

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