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Thread: Struggle to define the pmf for a specific random variable

  1. #1
    Senior Member dokrbb's Avatar
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    Struggle to define the pmf for a specific random variable

    I am working on the following question.

    We toss a coin $k$ times with the probability of getting heads in each toss being $p$.

    I also define a random variable
    $Y =
    \begin{cases}
    5^{k} & \text{if heads doesn't appear in the k tosses} \\ 5^{i} & \text{if first heads appear in the i-th toss for } 1 \leq i \leq k
    \end{cases}$

    Let's define each case as $Y_{1} - \text{not getting a heads on the first k tosses and } Y_{2} - \text{getting the first heads on the i-th toss}$

    Then, $p(y_{1_{i}} \in Y_{1}) = (1-p)^{k}, \text{whereas } p(y_{2_{i}} \in Y_{2}) = (1-p)^{i-1}p$, by following the Geometric distribution.

    Then when defining the pmf of $y_{i} \in Y, p(y_{i} \in Y) = p(y_{1_{i}} \in Y_{1}) + p(y_{2_{i}} \in Y_{2}) $ which gives me

    $p(y_{i} = 5^{i}) = (1-p)^{k} + (1-p)^{i-1}p$ which looks kind of wrong to me...

    Could someone point me in the right direction?
    Thanks a lot
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  2. #2
    MHF Contributor
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    Re: Struggle to define the pmf for a specific random variable

    as you noted

    $P[\text{heads first rolled on roll i}] = p(1-p)^{i-1},~1\leq i \leq k$

    $P[\text{heads never rolled}]=(1-p)^k$

    Now $Y=5^k$ is assigned both when the heads is never rolled, and when heads is rolled on the $kth$ try. So

    $P[Y=5^i] = \begin{cases}

    p(1-p)^{i-1} &1 \leq i < k \\

    p(1-p)^{k-1}+(1-p)^k &i = k

    \end{cases}$
    Thanks from dokrbb
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