Probability and Statistics Tutorial Class, Week 23

Question One §

Question §

In Glasgow, half of the days have some rain. The local weather forecaster is correct $\frac{2}{3}$ of the time, i.e., the probability that she has predicted rain on a rainy day, and the probability that she predicts no rain on a dry day, is both equal to $\frac{2}{3}$.

When rain is forecast, a Glaswegian lady takes her umbrella. When rain is not forecast, she takes it with a probability of $\frac{1}{3}$.

1. Draw a tree diagram - start by labelling whether it rains, then what the forecaster will predict, then finally what the Glaswegian lady does.
2. Check that you know what types of probability you are labelling on the diagram.
3. Calculate the probability that the lady has no umbrella, given that it rains.
4. Calculate the probability that she brings her umbrella, given that it doesn’t rain.

Solution §

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Question Two §

Question §

Microchips are made by three companies. $30\%$ are supplied by the firm $I$, $50\%$ by $II$, and $20\%$ by $III$.

The probabilities of $A=\text{a defect in a chip}$ are $P(A|I)=0.03$, $P(A|II)=0.04$, and $P(A|III)=0.01$.

If an unlabelled box of chips turns up, what are the probabilities that he box came from $I$, $II$, or $III$…

1. … if a random test shows a defective chip?
2. … if a random test shows a non-defective chip?
3. … if the first chip was defective, and a second chip was then also tested and found defective?

Solution §

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Question Three §

Question §

Find the cumulative distribution function of the following probability density function:

$$f_X(x)\{\begin{array}{ll}0,& x<0\\ \frac{1}{2},& 0\le x\le 2\\ 0,& x>2\end{array}$$

Solution §

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Question Four §

Question §

Confirm that the CDF you found, $F_X(x)$,

1. is monotonically increasing;
2. tends to zero as $x$ tends to $-\infty$;
3. tends to one as $x\to \infty$.

Check your solution by recovering the original $f_X(x)$ (below) by differentiating the $F_X(X)$ you found.

$$f_X(x)\{\begin{array}{ll}0,& x<0\\ \frac{1}{2},& 0\le x\le 2\\ 0,& x>2\end{array}$$

Solution §

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