A system for a random amount of time X (in units of months) is given by a density ;f(x)&nbsp;=&nbsp;14xe-x2&nbsp;;&nbsp;x&nbsp;&gt;&nbsp;0&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;0;&nbsp;x≤0(a) Find the moment generating function of X . Hence, compute the variance of X .&nbsp;(b) Deduce the expression for the k th moment.&nbsp;(c)Obtain the distribution function of X . Hence, compute that the probability that, 7 of such system, at least 4 will function for at least 6 units of months. State the assumptions that you make.&nbsp;

Question

A system for a random amount of time X (in units of months) is given by a density ;f(x)&amp;nbsp;=&amp;nbsp;14xe-x2&amp;nbsp;;&amp;nbsp;x&amp;nbsp;&amp;gt;&amp;nbsp;0&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;0;&amp;nbsp;x≤0(a) Find the moment generating function of X . Hence, compute the variance of X .&amp;nbsp;(b) Deduce the expression for the k th moment.&amp;nbsp;(c)Obtain the distribution function of X . Hence, compute that the probability that, 7 of such system, at least 4 will function for at least 6 units of months. State the assumptions that you make.&amp;nbsp;

Mr Solver · Accepted Answer

To solve the given problem, we&#039;ll follow these steps:(a) Find the moment generating function (MGF) of X and compute the variance of X.The moment generating function (MGF) of a random variable X is defined as the expected value of e^(tX), where t is a real parameter. It can be represented as:MX(t)=E[etX]To find the MGF of X, we need to calculate the expected value of e^(tX) using the given density function.MX(t)=E[etX]=∫−∞∞etxf(x)dxHowever, the given density function f(x) is only defined for x &amp;gt; 0. Therefore, we can rewrite the integral as follows:MX(t)=∫0∞etx(14xe−x2)dxMX(t)=14∫0∞e(t−12)xxdxNow, let&#039;s evaluate this integral. We&#039;ll use a property of the exponential integral, which states that:∫0∞e−sxxdx=ln(s),for&amp;nbsp;s&amp;gt;0Comparing this with our integral, we can see that the exponent inside the integral matches the form of the exponential integral property. Therefore, we can substitute s = t - 1/2 into the property:MX(t)=14ln(t−12)This is the moment generating function (MGF) of X.To compute the variance of X, we need to differentiate the MGF twice with respect to t and evaluate it at t = 0. The variance can be found using the following formula:Var(X)=MX″(0)−[MX′(0)]2Let&#039;s differentiate M_X(t) with respect to t:MX′(t)=14(t−12)Now, differentiate M_X&#039;(t) with respect to t:MX″(t)=−14(t−12)2Substituting t = 0 into these derivatives:MX′(0)=14(0−12)=−12MX″(0)=−14(0−12)2=−2Finally, we can compute the variance of X:Var(X)=MX″(0)−[MX′(0)]2=−2−(−12)2=−2−14=−94

A system for a random amount of time

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