Suppose that diameters of a shaft s manufactured by a certain machine are normal random variables with mean 10 and s.d. \(0.1 .\) If for a given application the shaft must meet the requirement that its diameter falls between \(9.9\) and \(10.2 \mathrm{~cm}\). What proportion of shafts made by this machine will meet the requirement?

Let \(X_{1}, X_{2}, \ldots\) be a sequence of independent and identically distributed random variables with mean 1 and variance 1600 , and assume that these variables are nonnegative. Let \(Y=\sum_{k=1}^{100} X_{k} .\) Use the central limit theorem to approximate the probability \(P(Y \geq 900)\).

A random number is chosen from the interval \([0,1]\) by a random mechanism. What is the probability that (i) its first decimal will be 3 (ii) its second decimal will be 3 (iii) its first two decimal will be 3 's?

An urn contains \(n\) cards numbered \(1,2, \ldots, n\). Let \(X\) be the least number on the card obtained when \(m\) cards are drawn without replacement from the urn. Find the probability distribution of random variable \(X\). Compute \(P(X \geq 3 / 2)\).

A bombing plane flies directly above a railroad track. Assume that if a larger(small) bomb falls within \(40(15)\) feet of the track, the track will be sufficiently damaged so that traffic will be disrupted. Let \(X\) denote the perpendicular distance from the track that a bomb falls. Assume that $$ f_{X}(x)=\left\\{\begin{array}{ll} \frac{100-x}{5000}, & \text { if } x \in 0

Let \(Y \sim N\left(\mu, \sigma^{2}\right)\) where \(\mu \in \mathbb{R}\) and \(\sigma^{2}<\infty .\) Let \(X\) be another random variable such that \(X=e^{Y}\). Find the distribution function of \(X\). Also, verify that \(E(\log (X))=\mu\) and \(\operatorname{Var}(\log (X))=\sigma^{2}\).