# Question: 1if the random variable y has an expected value of...

###### Question details

**1.**If the random variable Y has an expected value of μy and a variance of σy^2 , what are the expected value and variance of W=aY+b ?

**2**. Suppose that the random variable X~N(μ,σ^2). Then, show that Y=(X-μ)^2/σ^2 has a chisquare (X^2) distribution with one degree of freedom.

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**3.** Let X be the sample mean of a random sample of size 5 from a normal distribution with mean =0 and variance = 125. Determine c such that P(x̄<c)=0.90.

**4.** If x̄ is the sample mean of a random sample of size n from a normal distribution with mean =μ and variance = 100. Determine n so that P( μ-5< x̄ <μ+5)=0.954

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**5.** Suppose X1,X2,X3,...,X25 and Y1,Y2,Y3,...,Y25 are two independent random samples from two normal distributions N(0,16) and N(1,9), respectively. Let x̄ and **Ý **denote the corresponding sample means. Compute P(x̄> **Ý )**. (Consider the distribution of x̄ -**Ý **.)

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**6.** Let x̄ and S^2=∑(i=1,n)(Xi-x̄ )^2/n be the sample mean and sample variance of a random sample of size 25 from a distribution that is N(3,100). Evaluate P(0<x̄ <6 and 55.2<S^2<145.6).