Question

Assume that the distance at which Mr Jay throws a weight, X, be distributed as a Rayleigh variable with parameter σ=10m.

Answer that questions below, rounding the answers to 4 digits.

(a) What is the expected value of X?

(b) What is the variance X?

(c) Enter the formula of the quantile function F^(-1)(m), the function solving x: F(x)=m

(d) If the world record is 13.3m, what is the probability that Jay will break it?

(e) Give P(8<X<=12)

Answer 1

Answer:

Given Data

Given that throwing distance of a weight is x be distributed as a Rayleigh variable with parameter  σ=10m

Distribution is Rayleigh variable Distribution

So , probability density function is ,

Mean =

Variance =

Quantile

Now , we discuss given question

a)What is the expected value of X.

Expected value of x means mean of x

Expected value E(x) =

=

= 10*1.253

= 12.53

value is 3.14

b)  What is the variance X.

Variance of X =  

=

= 43

(c) Enter the formula of the quantile function F^(-1)(m), the function solving x: F(x)=m

Quantile function

   =

(d) If the world record is 13.3m, what is the probability that Jay will break it.

World record is x = 13.3 m

We have to find probability of x > 13.3

So world record is broken by Jay

So we use cumulative distribution function for find because it is used to find probability of x will take a value less or equal to x.

Cumulative distribution function of Rayleigh variable is

  

= 1 - 0.4129

= 0.5871

P(x > 13.3 ) = 1 - 0.5871

= 0.4129

Probability of breaking world record = 0.4129

(e) Give

Again use cumulative distribution function for above

  

= 1 - 0.487 - 1 + 0.726

= 0.239

   = 0.239

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