Find the probability

Each item produced by a certain manufacturer is, independantly, of acceptable quality with probability 0.95. Approximate the probability that at most 10 of the next 150 items produced are unacceptable.

Continuous Random Variables - Introductory Probability - 400 Level Class

An image is partitioned into 2 regions - one white and the other black. A reading taken from a randomly chosen point in the white section will give a reading that is normally distributed with μ=4 and σ²=4, whereas one taken from a randomly chosen point in the black region will have a normally distributed reading with parameters (6,9). A point is randomly chosen on the image and has a reading of 5. If the fraction of the image that is black is α, for what vlaue of α would the probability of making an error be the same whether on concluded the point was in the black region or in the white region? → NOTE: Black is N(6, 3) (9 = σ²)

Continuous Random Variables - Introductory Probability - 400 Level Class

a) A fire station is to be located along a road of length A, A<∞. If fires will occur at points uniformly chosen of (0, A), where should the station be located so as to minimize the expected distance from the fire? That is, choose a so as to minimize E[|X-a|] when X is uniformly distributed over (0,A). b) Now suppose that the road is of infinite length - stretching from point 0 outward to ∞. If the distance of a fire from point 0 is exponentially distributed with rate λ, where should the fire station now be located? That is, we want to minimize E[|X-a|], where X is now exponential with rate λ.

Continuous Random Variables - Introductory Probability - 400 Level Class

If X is uniformly distributed over (0, 1), find the destiny function of Y = e^x.

Continuous Random Variables - Introductory Probability - 400 Level Class

Verify that Var(X) =α/λ² when X is a gamma random variable with paramaters α and λ.