let (x) be a random variable whose pdf is given by the plot below. assume that the pdf is zero outside of…

Explanation:

Step1: Estimate the mean

The mean of a symmetric distribution (the given pdf looks approximately symmetric) is at the center of symmetry. From the plot, the center of the distribution seems to be around $x = 2$. So, we estimate the mean $\mu\approx2$.

Step2: Estimate the standard - deviation

For a normal - like distribution, about 68% of the data lies within one standard deviation of the mean, 95% within two standard deviations, and 99.7% within three standard deviations. Looking at the plot, the interval that contains about 95% of the data seems to be from approximately $x = 0$ to $x = 4$. If we assume this is a two - standard - deviation interval ($\mu - 2\sigma$ to $\mu+2\sigma$) and $\mu = 2$, then $4=2 + 2\sigma$ and $0=2-2\sigma$, which gives $\sigma\approx1$.

Step3: Maximize $P(a\leq X\leq a + 2)$

The probability $P(a\leq X\leq a + 2)$ is maximized when the interval $[a,a + 2]$ contains the most of the area under the pdf curve. Since the distribution is symmetric about $x = 2$, when $a=1$, the interval $[1,3]$ is centered around the peak of the distribution. So, $a = 1$ maximizes $P(a\leq X\leq a + 2)$.

Step4: Estimate $P(0\leq X\leq 2)$

We can estimate the area under the curve from $x = 0$ to $x = 2$ by visually comparing it to the total area under the curve (which is 1). The area from $x = 0$ to $x = 2$ seems to be about 0.35.

Answer:

Estimate the mean of $X$: 2 Estimate the standard deviation of $X$ to within 0.5: 1 For which $a$ is $P(a\leq X\leq a + 2)$ maximized: 1 Estimate $P(0\leq X\leq 2)$ to two decimal places: 0.35