Evaluate the probability P(0.5 2264 X 2264 1.5).

Question

ScanSolve AI Tutor · Accepted Answer

Here's the evaluation of the probability:

To evaluate the probability $P(0.5 \le X \le 1.5)$ , we use the Cumulative Distribution Function (CDF) $F(x) = \frac{3x^2}{4} - \frac{x^3}{4}$ . The probability is given by $P(0.5 \le X \le 1.5) = F(1.5) - F(0.5)$ .

Calculate $F(1.5)$ .
Substitute $x = 1.5$ (or $\frac{3}{2}$ ) into the CDF:
$F(1.5) = \frac{3(1.5)^2}{4} - \frac{(1.5)^3}{4}$ $F(1.5) = \frac{3\left(\frac{3}{2}\right)^2}{4} - \frac{\left(\frac{3}{2}\right)^3}{4}$ $F(1.5) = \frac{3\left(\frac{9}{4}\right)}{4} - \frac{\frac{27}{8}}{4}$ $F(1.5) = \frac{\frac{27}{4}}{4} - \frac{27}{32}$ $F(1.5) = \frac{27}{16} - \frac{27}{32}$
To subtract, find a common denominator, which is 32:
$F(1.5) = \frac{27 \times 2}{16 \times 2} - \frac{27}{32} = \frac{54}{32} - \frac{27}{32} = \frac{27}{32}$
Calculate $F(0.5)$ .
Substitute $x = 0.5$ (or $\frac{1}{2}$ ) into the CDF:
$F(0.5) = \frac{3(0.5)^2}{4} - \frac{(0.5)^3}{4}$ $F(0.5) = \frac{3\left(\frac{1}{2}\right)^2}{4} - \frac{\left(\frac{1}{2}\right)^3}{4}$ $F(0.5) = \frac{3\left(\frac{1}{4}\right)}{4} - \frac{\frac{1}{8}}{4}$ $F(0.5) = \frac{\frac{3}{4}}{4} - \frac{1}{32}$ $F(0.5) = \frac{3}{16} - \frac{1}{32}$
To subtract, find a common denominator, which is 32:
$F(0.5) = \frac{3 \times 2}{16 \times 2} - \frac{1}{32} = \frac{6}{32} - \frac{1}{32} = \frac{5}{32}$
Calculate $P(0.5 \le X \le 1.5)$ .
$P(0.5 \le X \le 1.5) = F(1.5) - F(0.5)$ $P(0.5 \le X \le 1.5) = \frac{27}{32} - \frac{5}{32}$ $P(0.5 \le X \le 1.5) = \frac{27 - 5}{32} = \frac{22}{32}$
Simplify the fraction
$P(0.5 \le X \le 1.5) = \frac{11}{16}$
The probability is $\frac{11}{16}$ .

Evaluate the probability P(0.5 2264 X 2264 1.5).

Answer

Step-by-step solution

Calculate $F(1.5)$ .

Calculate $F(0.5)$ .

Calculate $P(0.5 \le X \le 1.5)$ .

Simplify the fraction

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Evaluate the probability P(0.5 2264 X 2264 1.5).

Answer

Step-by-step solution

Calculate F(1.5)F(1.5)F(1.5).

Calculate F(0.5)F(0.5)F(0.5).

Calculate P(0.5≤X≤1.5)P(0.5 \le X \le 1.5)P(0.5≤X≤1.5).

Simplify the fraction

Try one yourself

Still stuck on this one?

More Mathematics Questions

Calculate $F(1.5)$ .

Calculate $F(0.5)$ .

Calculate $P(0.5 \le X \le 1.5)$ .