Prove the "memoryless property" of Exponential distribution and geometric distribution: let X ~ Exponential(), Y ~ Geometric(p). Then, P(X...
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the <span class="hljs-keyword">first</span> prat <span class="hljs-keyword">for</span> question <span class="hljs-number">7.</span> But i <span class="hljs-keyword">do</span> <span class="hljs-keyword">not</span> know how <span class="hljs-keyword">to</span> deal <span class="hljs-keyword">with</span> the <span class="hljs-keyword">second</span> part. Also, the proof <span class="hljs-keyword">of</span> question <span class="hljs-number">8</span> <span class="hljs-keyword">is</span> hard <span class="hljs-keyword">for</span> me <span class="hljs-keyword">as</span> well. </pre>

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7. Prove the &quot;memoryless property&quot; of Exponential distribution and geometric distribution: let
X ~ Exponential(), Y ~ Geometric(p). Then,
P(X &gt; s + t| X &gt; s) = P(X &gt; t), Vs, t&gt; 0;
(1)
P(Y-12k+m|Y -12 m) = P(Y &gt; k), V non-negative intergers k, m.
(2)
8. If a random variable Y only takes values in {1, 2, 3, ...} and satisfies (2), prove Y must satisfies
geometric distribution for some 0 &lt; p &lt; 1.
If a continuous random variable X only takes non-negative real values and satisfies (1), it
must be exponential distributed for some B &gt; 0.
Hint: First try to find out the value of P(X &gt; x) for all integers; then try to find out the value
of P(X 2 x) for all rational numbers x; finally, use the fact P(X &gt; x) is a non-increasing
function.

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