The projection X of the radius-vector of a random point on L a circumference of radius a onto the diameter has the distribution function (the arcsine law) F(x) { +arcsin for-a< x <6; Determine (a) the probability that X will be on the interval (-a/2, a/2), (b) the quantile x0.75, (c) the probability density f(x) of the random variable X, (d) the mode and median of the distribution.

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A:

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A: The cumulative distribution function (CDF) of the continuous random variable W is

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A:

Q: (a) P(X< 50) (c) P(40 < X< 60) (e) P(0<X<10) (b) P(X < 40) (d) P(X<0) (f) P(-10<X<10)

A:

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Q: Question is attached.

A: (a)The value of c can be derived as follows:

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A:

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A: a) Take k=1/4 Consider, ∫02146x-3x2dx=146x22-3x3302=143x2-x302=1412-8=44=1 Since ∫02fxdx=1,k = 1/4…

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A: We have   Cumulative distribution function for the probability density function.

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A: X and Y denote  jointly continuous rvs.The joint density function:

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A: 10. Note: The moment generating function for the normal distribution Nμ,σ2 is eμt+σ2t22........(1).…

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A: Solution

Q: How would

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