Can someone please let me know if my solutions for these problems are correct?
Suppose a pitcher in baseball is pitching to a batter, and the perfect pitch lands at the catcher's glove right at the "origin" of a coordinate system. That is, suppose the y-axis runs vertically from the ground to the sky and the x-axis runs horizontally from left to right from the pitcher’s vantage point. Suppose the batter is right-handed. Hence, a pitch thrown too far in the positive x direction would hit the batter.
Assume the pitcher’s fastball’s "x" coordinate is normally distributed with mean 0 (a perfect pitch) and variance $\sigma^2$. Suppose the batter is 1 foot away from the origin. Assume the pitcher always throws at the proper "height" (vertical direction).
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a) If the strike zone is 6 inches, find an expression for the probability the pitcher throws a strike in terms of $\Phi$. Plot this probability as a function of $\sigma^2$ (at least 5 plotted points).
$$P(-0.5 \leq X \leq 0.5) = P(\frac{-0.5}{\sigma} \leq Z \leq \frac{0.5}{\sigma})$$
$$=\Phi(\frac{0.5}{\sigma}) - \Phi (\frac{-0.5}{\sigma})$$
I made a table with values for $\sigma$ and the probability and just plotted it as follows:
b) Find an expression for the probability the batter gets hit, if the batter is 2 feet "wide" and 1 foot away from home plate, as a function of $\sigma^2$ (again in terms of $\Phi$). Calculate this probability for $\sigma^2 = 0.5$ and $\sigma^2 = 2$.
$$P(1\leq X \leq 3)=P(\frac{1}{\sigma}\leq Z \leq \frac{3}{\sigma})$$
$$=\Phi(\frac{3}{\sigma})-\Phi(\frac{1}{\sigma})$$
When $\sigma^2=0.5:$
$$\Phi(\frac{3}{\sqrt{0.5}})-\Phi(\frac{1}{\sqrt{0.5}})\approx 0.079$$
When $\sigma^2=2:$
$$\Phi(\frac{3}{\sqrt{2}})-\Phi(\frac{1}{\sqrt{2}})\approx 0.22$$
c) Ignoring the fact that the pitcher always pitches at the correct height, how does this model poorly model this physical situation?
The pitcher being right-handed skews the distribution to left since they're trying to not hit the batter.
d) How does this model accurately model this situation?
It shows that the closer you pitch to the center, it's more probable you land in the glove.
e) What does a smaller $\sigma^2$ indicate in this context?
Smaller $\sigma^2$ means they're a better pitcher since the probability of the pitch getting farther from the mean decreases.